It is known that not every convex polytope (even polyhedron, e.g. [this one](https://en.wikipedia.org/wiki/Triakis_tetrahedron)) can be made inscribed, that is, we cannot always move its vertices so that

 - all vertices end up on a common sphere, and
 - the polytope has not changed its combinatorial type in the process.

Is there anything known about whether this is possible if we instead ask for [*simple polytopes*](https://en.wikipedia.org/wiki/Simple_polytope), i.e., $d$-dimensional polytopes of vertex-degree $d$?