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](https://arxiv.org/abs/math/9812033)), but I have not enough understanding of this construction yet.