Is there a neat way to show (or a reference that already proves) that

- the 4-cube is the only convex 4-polytope in which all facets are
*regular*3-cubes? - the 24-cell is the only convex 4-polytope in which all facets are
*regular*octahedra? - the 120-cell is the only convex 4-polytope in which all facets are
*regular*dodecahedra?

Note that I do require that the facets are *regular*, so e.g. just any cubical polytope does not fit.

I am aware of Gosset's semiregular polytopes (vertex-transitive and all facets are regular polytopes), for which this statement is true, but I do not require vertex-transitivitiy here.

However, if my statement turns out to be wrong in this general form, then I wonder whether it holds when I require that all vertices are on a common sphere.