These few days I've been trying to tackle the problem of non-EKF crown jewels again. It seems that lacking any firm methodology as to how to search for them, and lacking a concrete algorithm for brute-force searching, the next best thing might be to implement a "4D CRF lego" software that lets you build the polytopes piece-by-piece until you find something that closes up.

Another line of thought I have is with the snub antiprism sequence in 3D: digonal snub antiprism = snub disphenoid; trigonal snub antiprism = icosahedron; snub square antiprism; and snub pentagonal antiprism (concave, non-CRF). These all start with some base polygon, with attached triangles ("edge pyramids"), and forming some kind of skew polygon where the other half is joined. In terms of higher dimensions, it seems a better way to see the other half is not merely some rotation of the first half, flipped, but rather as a similar construction of triangles ("edge pyramids") surrounding the dual of the starting base polygon. So in 4D, potentially we could look in the direction of some base polyhedron P, surrounded by some "skirt" of pyramids of appropriate base shapes to fit onto the faces of P, then the dual of P, also with its own set of pyramids, and try to find a configuration where the skew polygon around the unattached boundary of the "pyramid skirts" have the same shape, and therefore can be joined together to form a closed polychoron. (There might also be some other pyramids like tetrahedra to serve as lacing cells that may not directly share a face with P or its dual.) To help with finding more candidates, we could perhaps relax the CRF requirement at first, and study families of such polytopes, to see if we can find what are the conditions for their existence, and afterwards evaluate which ones among them might possibly be CRF-able.