Varchenko–Stasinski's conjecture on odd-length generating functions for type B parabolic quotients

Let $n\in {\mathbb P}$ and $J\subseteq [0,n-1]$. Let $J_0$ be the possibly empty connected component of $J$ containing $0$, and let $J_1,\ldots,J_s$ be the remaining connected components. Set

$$\widetilde{m}:=\sum_{i=1}^{s}\left\lfloor\frac{|J_i|+1}{2}\right\rfloor,$$

and let $a:=\min([0,n]\setminus J)$. Let $B_n^J$ be the relevant parabolic quotient in the hyperoctahedral group, $\ell$ the Coxeter length, and $L$ the type-$B$ odd-length statistic.

Varchenko–Stasinski's conjecture. The generating function satisfies

$$\sum_{\sigma\in B_n^J}(-1)^{\ell(\sigma)}x^{L(\sigma)}=\frac{\prod_{j=a+1}^{n}(1-x^j)}{\prod_{i=1}^{\widetilde{m}}(1-x^{2i})}\left[\begin{array}{c}\widetilde{m}\\\\\left\lfloor\frac{|J_1|+1}{2}\right\rfloor,\ldots,\left\lfloor\frac{|J_s|+1}{2}\right\rfloor\end{array}\right]_{x^2}.$$

The paper states that this conjecture is proved as a consequence of its results, so it is solved.