Cesi's spectral-gap conjecture for the hyperoctahedral group

Let $B_n=C_2\wr S_n$ be the hyperoctahedral group, with $S_n$ realized as permutation matrices and, for each subset $A\subseteq\{1,\dotsc,n\}$, let $s_A$ be the diagonal signed permutation having diagonal entries $-1$ exactly at the positions in $A$. Let $P_n$ be the $2n$-dimensional representation whose irreducible components correspond to $([n-1,1],\emptyset)$, $([n-1],[1])$, and $([n],\emptyset)$. Suppose that $w=w_N^-+w_T\in\mathbb{R}[B_n]$, where $w_T$ is a nonnegative combination of transpositions $(ij)\in S_n$ and $w_N^-$ is a nonnegative combination of the elements $s_A$ for subsets $A\subseteq\{1,\dotsc,n\}$ of odd size.

Cesi's conjecture.

$$\psi_{B_n}(w)=\psi_{B_n}(w,P_n).$$

Cesi proposed this restricted extension of his spectral-gap theorem after showing that the analogous assertion with all diagonal elements does not hold in general for the representation $P_n$. The supplied text gives no resolution of this conjecture.