In An enduring error, Branko Grünbaum discusses the question of defining the Archimedean solids, noting the case of the 'pseudo-rhombicuboctahedron', sometimes called Miller's solid and shown on the right above, as opposed to the rhombicuboctahedron, one of the standard $13$ Archimedean solids and shown on the left (the difference being that the triangles and squares in the lower portion of the body have been 'shifted around' by one place). What has caused an 'enduring error' is a failure to distinguish properly between two criteria, namely:
Local criterion: All faces are regular polygons, and the cyclic arrangement of the faces around each vertex is the same.
and
Global criterion: All faces are regular polygons, and all vertices form one orbit under isometric symmetries of the polyhedron.
The rhombicuboctahedron satisfies both the local and global criterion, while the pseudo-rhombicuboctahedron only satisfies the local criterion.
At the end of the article, Grünbaum indicates that whether there are any other such solids, i.e. convex solids which satisfy the local but not the global criteria, is an open question, conjecturing that there aren't any such solids but that 'a proof of this is probably quite complicated.'
My question is that, given that Grünbaum's paper was written in 2009, does anyone know if any progress has been made in determining whether there are any such solids (Wikipedia indicates that there hasn't)? Assuming there hasn't been, I thought it would be interesting to hear any thoughts on why it's so difficult to show that there aren't any further such solids.
(I note that if we allow self-intersecting polyhedra then there is the pseudo-great rhombicuboctahedron, corresponding to the non-convex great rhombicuboctahedron.)
Given the definitions of local and global uniformity, the problem has long been solved – at least for the convex case.
The Johnson solids are the strictly convex polyhedra with regular polygon faces that are not vertex-transitive. Zalgaller proved in 1969 that there are only $92$ of them, and only Miller's solid out of them satisfies the local but not the global uniformity criterion.