Does there exist a convex polytope $P\subset \Bbb R^d,d\ge 3$, other than the $d$-cube, so that

• $P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
• all 2-faces of $P$ are 4-gons (not necessarily squares, or rectangles).

I expect such a polytope, if at all, then only in $d\ge 4$. Maybe one of the neighborly cubical polytopes constructed by Ziegler (see here), but I have not enough understanding of this construction yet.

What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds

• $P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
• all 2-faces of $P$ are 4-gons (not necessarily squares, or rectangles).

I know the $d$-cubes, rhombic dodecahedron and rhombic triacontahedron. Are there any others?

I expect other such a polytope, if at all, then only in $d\ge 4$. Maybe a slight modification of a neighborly cubical polytope as constructed by Ziegler here, but as they are, they have two 2-face orbits.