The simplicity conjecture for convex hulls of signed Wachs permutations

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

This conjecture is part of the paper's proposed program on convex hulls of Wachs and signed Wachs permutations. The supplied text gives no verification range or resolution for the signed statement, so it remains open.