Today I stumbled across Steffen's polyhedron, which is a non-convex, closed polyhedron that has flexible dihedral angles in spite of being a closed shape. (It is proven that no convex polyhedron can be flexible.)

It makes me wonder, is it possible for a non-convex 4D polytope to have flexible vertices, in spite of Marek's proof? Or is this only possible in 3D? An initial idea I have is to assemble a polychoron out of Steffen's polyhedra. Since the dihedral angles of these cells would then be flexible, the vertex figure involving such a polyhedron also ought to be flexible. Not sure if such a shape can be closed up into a polychoron while maintaining flexibility, though!