Are (quasi-)regular polytopes uniquely determined by their edge graph? I consider polytopes $P\subset\Bbb R^n,n\ge 2$ of arbitrary dimension (intersection of finitely many halfspaces, therefore convex), which are vertex- and edge-transitive (also called quasi-regular). 

Question: Can there exist two different such polytopes $P_1\subset \Bbb R^{n_1}$ and $P_2\subset\Bbb R^{n_2}$, maybe even of different dimensions, which have isomorphic edge-graphs (1-skeletons)? If one polytope is just the other one but embedded in a higher dimension, I consider these to be the same.

What if I drop edge-transitivity and instead use some suitable higher-dimensional generalization of uniform polyhedrons, or even weaker, require only that the edges are of the same length.

Update
I found two statements, relevant for this question:


*

*Simple polytopes are uniquely determined by their edge-graphs. However, the definition of "simple polytope" fixes the dimension, so there might be higher dimensional realizations too.

*This answer on MO (and the comments) explain that for $K_n,n\ge 5$, there is a polytope of dimension $4\le d\le n-1$ which has $K_n$ as an edge-graph (see neighborly polytopes). So the dimension is not uniquely determined. However, I do not know which of these realizations of $K_n$ is vertex- and/or edge-transitive.


During my research, I found out that only vertex-transitivity and convexity is not enough to ensure that the 1-skeleton is uniquely realizable. So the remaining question is whether edge-transitivity, uniformity or a requirement on the edge lengths can make it unique.
The counter-example I found was among the cyclic polytopes, which are a specific way to construct neighborly polytopes in higher dimensions. E.g. the $K_n,n\ge 5$ can be realized as a simplex in $n-1$ dimensions, or as a cyclic polytope in a lower dimension. 
The crucial detail I now came across was that cyclic polytopes in even dimensions are combinatorially vertex-transitive and can be realized in the same dimension in a way that realizes all the automorphisms of their face-lattice (see this paper). This means: besides the simplices, there are indeed vertex-transitive (convex) polytopes with $K_n$ as their 1-skeleton. However, I found nothing about whether they can be realized edge-transitive or with any other symmetry.

Update
I just realized, that the only way to place $n$ points equidistant, we need to arrange them as the vertices of the $(n-1)$-dimensional simplex.  Therefore, the only realization of $K_n$ as an edge-transitive polytope (or even as a polytope with equal edge lengths) is the simplex.
Hence, a possible counter-example cannot be neighborly.
