Orthogonal Igusa-function polynomial conjecture

Let ${\mathcal{V}}$ be an $n$-dimensional, non-degenerate quadratic space, let $[n-1]=\{1,\ldots,n-1\}$, let ${\mathcal{C}}_n$ be the subgroup of chessboard elements in the symmetric group ${\mathcal{S}}_n$, let $D_{\operatorname{L}}(w)$ be the left-descent set of $w$, let $L$ be the specified linear combination of parabolic length functions, and let $\chi_{\varepsilon}$ be the specified linear character of ${\mathcal{C}}_n$. Orthogonal Igusa-function polynomial conjecture. For each $J\subseteq[n-1]$, the polynomial associated with ${\mathcal{V}}$ satisfies

$$\alpha^J_{{\mathcal{V}}}(q^{-1})=\sum_{\substack{w\in{\mathcal{C}}_n\D_{\operatorname{L}}(w)\subseteq J}}\chi_{\varepsilon}(w)q^{-L(w)}.$$

If true, this gives the orthogonal case of the paper's main theorem from the corresponding general theorem and is a first step toward extending the other stated theorem to the orthogonal case; the parser supplies no evidence that the conjecture has been resolved.