The simplicity conjecture for convex hulls of Wachs permutations

Let $m\in\mathbb{P}$, and let $c(\operatorname{\mathcal{W}}(S_{2m}))$ be the convex hull of the Wachs permutations in $\mathbb{R}^{2m}$. Simplicity conjecture. The polytope $c(\operatorname{\mathcal{W}}(S_{2m}))$ is simple.

The authors verified this conjecture for $m\leqslant 5$, but report that SageMath shows $c(\operatorname{\mathcal{W}}(S_9))$ is not simple; this computational observation concerns the odd-dimensional analogue and does not resolve the stated even-index conjecture.