Let's say $P\subset\Bbb R^d$ is some *convex* polytope with the following two properties:

- all vertices are on a common sphere.
- all edges are of the same length.

I suspect that such a polytope is already rigid, i.e. there is (up to scaling and rotation) only a single way to realize it geometrically. Is this true? What if we only look at uniform polytopes?