Seems that diminished icositetrachora ARE part of augmented tesseracts, EXCEPT:
Icositetrachoron has 24 vertices, while tesseract only has 8 cells. Augmented tesseracts correspond to icositetrachora with vertices cut off from a specific 8-vertex subset. But it should be possible to cut off two or more vertices from DIFFERENT subsets, if they are far enough from each other - and those shapes wouldn't correspond to any augmented tesseract.
I think that the search for Johnsonian polychora needs to be two-fold:
1. Search for the primitive polychora, i.e. those that can't be diminished.
2. Put them together in various convex ways (bearing in mind that, unlike 3D case, here it's possible for two cells to be cohyperplanar if it means they will create a larger cell.
I'd suggest starting by enumerating the possible configurations of vertices and edges.
Let's look at possible dihedral angles first (all numbers in degrees). I assume that heptagonal, nonagonal and 11+-gonal prisms and antiprisms can't form anything more than duoprisms and antiprismatic prisms (although... some of those might be possible to augment?), so I omit those. Angles were taken from Great Stella.
So far it contains dihedral angles up to gyroelongated square dipyramid, feel free to extend
31,7175 - 4-10 angle of pentagonal cupola
37.3774 - 3-5 angle of pentagonal pyramid
3-10 angle of pentagonal cupola
Relations: 138.19 - 100.812, 127.377 - 90, half of 74.7547
45 - 4-8 angle of square cupola
Relations: half of 90, third of 135, 135 - 90
54.7356 - 3-4 angle of square pyramid
4-6 angle of triangular cupola
3-8 angle of square cupola
Relations: half of 109.471; 125.264 - 70.5288, 144.736 - 90, 158.572 - 103.836
60 - 4-4 angle of triangular prism, 4-4 angle of elongated triangular pyramid, 4-4 angle of elongated triangular dipyramid
Relations: Complementary to 120, half of 120
63.4349 - 5-10 angle of pentagonal rotunda
Relations: 142.623 - 79.1877
70.5288 - the dihedral angle of tetrahedron, 3-3 angle of elongated triangular pyramid, axial angle of triangular dipyramid, 3-3 angle of elongated triangular dipyramid
6-6 angle of truncated tetrahedron
3-6 angle of triangular cupola
Relations: Complementary to 109.471; 125.264 - 54.7356, 160.529 - 90, half of 141.058
74.7547 - equatorial angle of pentagonal dipyramid
Relations: double of 37.3774
79.1877 - 3-10 angle of pentagonal rotunda
Relations: 142.623 - 63.4349
90 - the dihedral angle of cube, 4-4 angle of elongated square pyramid, 4-4 angle of elongated square dipyramid
8-8 angle of truncated cube
3-4 angle of triangular prism, base 3-4 angle of elongated triangular pyramid
4-5 angle of pentagonal prism, 4-5 angle of elongated pentagonal pyramid
4-6 angle of hexagonal prism
4-8 angle of octagonal prism
4-10 angle of decagonal prism
Relations: double of 45, 135 - 45, 144.736 - 54.7356, 160.529 - 70.5288, 127.337 - 37.3774
95.2466 - 3-10 angle of decagonal antiprism
96.5945 - 3-8 angle of octagonal antiprism
98.8994 - 3-6 angle of hexagonal antiprism
100.812 - 3-5 angle of pentagonal antiprism, 3-5 angle of gyroelongated pentagonal pyramid
Relations: 138.19 - 37.3774
103.846 - 3-4 angle of square antiprism, 3-4 angle of gyroelongated square pyramid
Relations: 158.572 - 54.7356
108 - 4-4 angle of pentagonal prism, 4-4 angle of elongated pentagonal pyramid
109.471 - dihedral angle of octahedron, 3-3 angle of square pyramid, 3-3 angle of elongated square pyramid, pyramid 3-3 angle of gyroelongated square pyramid, 3-3 angle of elongated square dipyramid, pyramidal angle of gyroelongated square dipyramid
3-6 angle of truncated tetrahedron
6-6 angle of truncated octahedron
Relations: Complementary to 70.5288, double of 54.7356
116.565 - dihedral angle of dodecahedron, 10-10 angle of truncated dodecahedron
120 - 4-4 angle of hexagonal prism
Relations: Complementary to 60, double of 60
125.264 - dihedral angle of cuboctahedron, 3-4 angle of triangular cupola
4-6 angle of truncated octahedron
3-8 angle of truncated cube
6-8 angle of truncated cuboctahedron
Relations: 54.7356 + 70.5288
127.377 - 3-4 angle of elongated pentagonal pyramid
Relations: 37.3774 + 90
127.552 - 3-3 angle of square antiprism, equatorial 3-3 angle of gyroelongated square pyramid, antiprismatic angle of gyroelongated square dipyramid
135 - 4-4 angle of rhombicuboctahedron, 4-4 angle of octagonal prism, 4-4 angle of square cupola
4-8 angle of truncated cuboctahedron
Relations: triple of 45, 45 + 90
138.19 - dihedral angle of icosahedron, 3-3 angle of pentagonal antiprism, 3-3 angle of pentagonal pyramid, 3-3 angle of elongated pentagonal pyramid, 3-3 angle of gyroelongated pentagonal pyramid, axial angle of pentagonal dipyramid
6-6 angle of truncated icosahedron
Relations: 37.3774 + 100.812
141,058 - equatorial angle of triangular dipyramid
Relations: double of 70,5288
142.623 - dihedral angle of icosidodecahedron, 3-5 angle of pentagonal rotunda
5-6 angle of truncated icosahedron
3-10 angle of truncated dodecahedron
6-10 angle of truncated icosidodecahedron
Relations: 63.4349 + 79.1877
142.983 - 3-4 angle of snub cube
144 - 4-4 angle of decagonal prism
144.736 - 3-4 angle of rhombicuboctahedron, 3-4 angle of square cupola, 3-4 angle of elongated square pyramid, 3-4 angle of elongated square dipyramid
4-6 angle of truncated cuboctahedron
Relations: 54.7356 + 90
145.222 - 3-3 angle of hexagonal antiprism
148.283 - 4-5 angle of rhombicosidodecahedron, 4-5 angle of pentagonal cupola
4-10 angle of truncated icosidodecahedron
152.93 - 3-5 angle of snub dodecahedron
153.235 - 3-3 angle of snub cube
153.962 - 3-3 angle of octagonal antiprism
158.572 - join 3-3 angle of gyroelongated square pyramid, join 3-3 angle of gyroelongated square dipyramid
Relations: 54.7356 + 103.836
159.095 - 3-4 angle of rhombicosidodecahedron, 3-4 angle of pentagonal cupola
4-6 angle of truncated icosidodecahedron
159.187 - 3-3 angle of decagonal antiprism
160.523 - side 3-4 angle of elongated triangular cupola
Relations: +
160.529 - apex 3-4 angle of elongated triangular pyramid, 3-4 angle of elongated triangular dipyramid
Relations: 70.5288 + 90
164.172 - 3-3 angle of snub dodecahedron
Vertex data (and these are complete):
3-3-3: tetrahedron, elongated triangular pyramid, triangular dipyramid, elongated triangular dipyramid, augmented tridiminished icosahedron
3-3-4: square pyramid
3-3-5: pentagonal pyramid
3-4-4: triangular prism, elongated triangular pyramid, gyrobifastigium, augmented triangular prism
3-4-6: triangular cupola
3-4-8: square cupola
3-4-10: pentagonal cupola
3-5-5: metabidiminished icosahedron, tridiminished icosahedron, augmented tridiminished icosahedron, bilunabirotunda
3-5-10: pentagonal rotunda
3-6-6: truncated tetrahedron, augmented truncated tetrahedron
3-8-8: truncated cube, augmented truncated cube, biaugmented truncated cube
3-10-10: truncated dodecahedron, augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron
4-4-4: cube, elongated square pyramid
4-4-5: pentagonal prism, elongated pentagonal pyramid, augmented pentagonal prism, biaugmented pentagonal prism
4-4-6: hexagonal prism, elongated triangular cupola, augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism
4-4-8: octagonal prism, elongated square cupola
4-4-10: decagonal prism, elongated pentagonal cupola, elongated pentagonal rotunda
4-5-10: diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, parabidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron
4-6-6: truncated octahedron
4-6-8: truncated cuboctahedron
4-6-10: truncated icosidodecahedron
5-5-5: dodecahedron, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron
5-6-6: truncated icosahedron
3-3-3-3: octahedron, square pyramid, elongated square pyramid, gyroelongated square pyramid, triangular dipyramid, pentagonal dipyramid, elongated square dipyramid, gyroelongated square dipyramid, augmented triangular prism, biaugmented triangular prism, triaugmented triangular prism, augmented pentagonal prism, biaugmented pentagonal prism, augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism, triaugmented hexagonal prism, snub disphenoid, augmented sphenocorona, sphenomegacorona
3-3-3-4: square antiprism, gyroelongated square pyramid, augmented triangular prism, biaugmented triangular prism, sphenocorona, augmented sphenocorona
3-3-3-5: pentagonal antiprism, gyroelongated pentagonal pyramid, metabidiminished icosahedron, tridiminished icosahedron, augmented tridiminished icosahedron, triangular hebesphenorotunda
3-3-3-6: hexagonal antiprism, gyroelongated triangular cupola
3-3-3-8: octagonal antiprism, gyroelongated square cupola
3-3-3-10: decagonal antiprism, gyroelongated pentagonal cupola, gyroelongated pentagonal rotunda
3-3-4-4: elongated triangular pyramid, elongated square pyramid, elongated pentagonal pyramid, elongated triangular dipyramid, elongated square dipyramid, elongated pentagonal dipyramid, triangular orthobicupola, square orthobicupola, pentagonal orthobicupola, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-4-5: pentagonal gyrocupolarotunda, augmented pentagonal prism, biaugmented pentagonal prism
3-3-4-6: augmented hexagonal prism, parabiaugmented hexagonal prism, metabiaugmented hexagonal prism, triaugmented hexagonal prism, triangular hebesphenorotunda
3-3-5-5: pentagonal orthobirotunda, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron, augmented tridiminished icosahedron
3-4-3-4: cuboctahedron, triangular cupola, elongated triangular cupola, gyroelongated triangular cupola, gyrobifastigium, triangular orthobicupola, square gyrobicupola, pentagonal gyrobicupola, elongated triangular orthobicupola, elongated triangular gyrobicupola, gyroelongated triangular bicupola, augmented truncated tetrahedron
3-4-3-5: pentagonal orthocupolarotunda, bilunabirotunda, triangular hebesphenorotunda
3-4-3-6: augmented truncated tetrahedron
3-4-3-8: augmented truncated cube, biaugmented truncated cube
3-4-3-10: augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron
3-4-4-4: rhombicuboctahedron, square cupola, elongated triangular cupola, elongated square cupola, elongated pentagonal cupola, gyroelongated square cupola, square orthobicupola, square gyrobicupola, elongated triangular orthobicupola, elongated triangular gyrobicupola, elongated square gyrobicupola, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, gyroelongated square bicopula, augmented truncated cube, biaugmented truncated cube
3-4-4-5: elongated pentagonal rotunda, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, elongated pentagonal orthobirotunda, elongated pentagonal gyrobirotunda, gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron
3-4-5-4: rhombicosidodecahedron, pentagonal cupola, elongated pentagonal cupola, gyroelongated pentagonal cupola, pentagonal orthobicupola, pentagonal gyrobicupola, pentagonal orthocupolarotunda, pentagonal gyrocupolarotunda, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda, augmented truncated dodecahedron, parabiaugmented truncated dodecahedron, metabiaugmented truncated dodecahedron, triaugmented truncated dodecahedron, gyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, diminished rhombicosidodecahedron, paragyrate diminished rhombicosidodecahedron, metagyrate diminished rhombicosidodecahedron, bigyrate diminished rhombicosidodecahedron, parabidiminished rhombicosidodecahedron, metabidiminished rhombicosidodecahedron, gyrate bidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron
3-5-3-5: icosidodecahedron, pentagonal rotunda, elongated pentagonal rotunda, gyroelongated pentagonal rotunda, pentagonal orthocupolarotunda, pentagonal gyrocupolarotunda, pentagonal orthobirotunda, elongated pentagonal orthocupolarotunda, elongated pentagonal gyrocupolarotunda, elongated pentagonal orthobirotunda, elongated pentagonal gyrobirotunda, gyroelongated pentagonal cupolarotunda, gyroelongated pentagonal birotunda, bilunabirotunda, triangular hebesphenorotunda
3-3-3-3-3: icosahedron, pentagonal pyramid, elongated pentagonal pyramid, gyroelongated square pyramid, gyroelongated pentagonal pyramid, pentagonal dipyramid, elongated pentagonal dipyramid, gyroelongated square dipyramid, biaugmented triangular prism, triaugmented triangular prism, augmented dodecahedron, parabiaugmented dodecahedron, metabiaugmented dodecahedron, triaugmented dodecahedron, metabidiminished icosahedron, snub disphenoid, snub square antiprism, sphenocorona, augmented sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-3-3-4: snub cube, gyroelongated triangular cupola, gyroelongated square cupola, gyroelongated pentagonal cupola, gyroelongated triangular bicupola, gyroelongated square bicopula, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda, snub square antiprism, augmented sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum
3-3-3-3-5: snub dodecahedron, gyroelongated pentagonal rotunda, gyroelongated pentagonal cupolarotunda, gyroelongated pentagonal birotunda