Stasinski–Voll's signed $L$-generating-function conjecture for hyperoctahedral descent classes

Let $B_n$ be the hyperoctahedral group, with Coxeter length $l:B_n\to\mathbb{N}_0$ and descent set

$$D(w)=\{i\in[n-1]_0\mid l(ws_i)<l(w)\}.$$

For $I=\{i_1,\dots,i_l\}_{<}\subseteq[n-1]_0$, put $i_{l+1}=n$ and define

$$B_n^{I^{\mathrm{c}}}=\{w\in B_n\mid D(w)\subseteq I\},$$

where $I^{\mathrm{c}}=[n-1]_0\setminus I$. For $w\in B_n$, let

$$L(w)=\frac{1}{2}\#\{(i,j)\in[\pm n]_0^2\mid i<j,\ w(i)>w(j),\ i\not\equiv j\bmod(2)\}.$$

Writing $(\underline{m})=1-X^m$, $(\underline{0})=1$, and

$$f_{n,I}(X)=\frac{(\underline{n})!}{(\underline{i_1})!}\prod_{r=1}^{l}\prod_{\sigma=1}^{\lfloor(i_{r+1}-i_r)/2\rfloor}(\underline{2\sigma})^{-1},$$

Stasinski–Voll's conjecture. For every $n\in\mathbb{N}$ and every $I=\{i_1,\dots,i_l\}_{<}\subseteq[n-1]_0$,

$$\sum_{w\in B_n^{I^{\mathrm{c}}}}(-1)^{l(w)}X^{L(w)}=f_{n,I}(X).$$

In particular, when $I=[n-1]_0$, this asserts that the signed $L$-generating function over all of $B_n$ equals $(\underline{n})!$. The conjecture gives explicit generating functions for the new statistic $L$ on hyperoctahedral descent classes; its resolution status is not specified in the supplied text.