Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
- All facets are congruent,
- it has an insphere (a sphere to which each facet is tangent to), and
- it is not facet-transitive.
In 3-dimensional space there is an example with the "memorable" name Pseudo-deltoidal icositetrahedron, depicted below. I believe its the only such polyhedron. I am not aware of any higher dimensional examples.