The twisting conjecture for quadrangulation flip graphs and simplicial complexes

Let $Q$ and $Q'$ be quadrangulations related by twisting along one inner edge. Write $\gamma_Q$ and $\gamma_{Q'}$ for their undirected flip graphs, and $G_Q$ and $G_{Q'}$ for their associated simplicial complexes. Twisting conjecture. The undirected flip graphs $\gamma_Q$ and $\gamma_{Q'}$ are isomorphic, and the simplicial complexes $G_Q$ and $G_{Q'}$ are isomorphic. The oriented flip-graphs are not asserted to be the same under this hypothesis. In particular, the conjecture predicts equality of the numbers of $Q$-compatible and $Q'$-compatible quadrangulations; a stronger enumerative invariance is proposed separately.