Noncanonical dual-graph conjecture for generalized snake order polytopes

Let $Q_{\mathbf{w}}$ be a poset with Jordan--Hölder poset $J(Q_{\mathbf{w}})=\widehat{P}(\mathbf{w})$, and let a turn mean the specified local turn in the generalized snake poset. A regular triangulation has a dual graph whose vertices are maximal simplices and whose edges join simplices sharing a facet. Noncanonical dual-graph conjecture. If $J(Q_{\mathbf{w}})=\widehat{P}(\mathbf{w})$ contains a turn, then $\mathcal{O}(Q_{\mathbf{w}})$ has a regular triangulation whose dual graph is not isomorphic to the dual graph of the canonical triangulation. The claim is supported by computations, while the ladder case has the same dual graph for every triangulation.