The Möbius-function conjecture for signed Wachs permutations

Let $n\in\mathbb{P}$, and let $\mu(u,v)$ denote the Möbius function of the Bruhat-order interval $[u,v]$ in the poset of signed Wachs permutations $\operatorname{\mathcal{W}}(B_n)$. Möbius-function conjecture. For all $u,v\in\operatorname{\mathcal{W}}(B_n)$,

$$\mu(u,v)\in\{0,1,-1\}.$$

The authors verified the conjecture for $n\leqslant 6$. They also note that the corresponding statement for ordinary Wachs permutations follows from this one through the stated poset interval isomorphism.