Regularity conjecture for the flip graph of generalized snake order polytopes

Let $\mathbf{w} in \mathcal{V}$, and consider the flip graph whose vertices are the regular triangulations of the order polytope $\mathcal{O}(Q_{\mathbf{w}})$. Let $k$ denote the dimension of its secondary polytope. Regularity conjecture. The flip graph of regular triangulations for $\mathcal{O}(Q_{\mathbf{w}})$ is $k$-regular. The preceding ladder computations and computational evidence motivate this assertion; its general validity is left open.