Nearly-ladder dual-graph counting conjecture for order polytopes

Assume $J(Q_{\mathbf{w}})=\widehat{P}(\mathbf{w})$, where $\mathbf{w}=\varepsilon L R^{n-2}$ and $ngeq 3$. Consider triangulations of $\mathcal{O}(Q_{\mathbf{w}})$ and their dual graphs, and compare them with the dual graph of the canonical triangulation. Nearly-ladder counting conjecture. The number of triangulations whose dual graph is isomorphic to the dual graph of the canonical triangulation is $4n(n-2)!$. The paper reports verification for $n=3,4,5,6,7$; the assertion is otherwise open.