Regularity conjecture for all triangulations of generalized snake order polytopes

Let $\mathbf{w} in \mathcal{V}$, and let $\mathcal{O}(Q_{\mathbf{w}})$ be the associated order polytope. A triangulation is regular if it arises from a lifting function, equivalently is a regular subdivision triangulation. All-triangulations regularity conjecture. All triangulations of $\mathcal{O}(Q_{\mathbf{w}})$ are regular. The paper proves that all such triangulations are unimodular and reports computational support for regularity; the general statement remains open.