Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$. The central symmetry of $P$ induces an involutory automorphism $\iota_P\in\mathrm{Aut}(G)$ of the edge-graph, and likewise $\iota_Q\in\mathrm{Aut}(G)$ for $Q$.

Question: do we have $\iota_P=\iota_Q$?

Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$. The central symmetry of $P$ induces an involutory automorphism $\iota_P\in\mathrm{Aut}(G)$ of the edge-graph, and likewise $\iota_Q\in\mathrm{Aut}(G)$ for $Q$.

Question: do we have $\iota_P=\iota_Q$?

It was pointed out by David E Speyer in the comments that this question can be interpreted in two different (yet still very related) ways. I admit, I don't know which version is the "right one" to ask for my application, and a solution to either would be welcome. I suspect that one of the versions is stronger, and this would then be the favorable one. But I also suspect that they are equivalent, in the sense that the answer is No to both.