Given a *finite* set of *convex* $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.

> **Question:** Is it true that there are only *finitely* many *convex* $(d+1)$-dimensional polytopes whose facets are solely (uniformly scaled and rotated versions of) polytopes in $\mathcal P$?

For example, there are only finitely many polyhedra that can be built from any finite set of regular polygons, but as far as I know, this result is by enumeration  (see, e.g. [Johnson solids](https://en.wikipedia.org/wiki/Johnson_solid)).