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).