The parking-number identity for arbitrary finite Coxeter systems

Let $(W,S)$ be an irreducible finite Coxeter system with Coxeter number $h$, let $\vec{c}$ be a Coxeter word in $S$, let $p>0$ be an integer coprime to $h$, and let $J\subseteq S$. Define the rational parabolic $\X$-parking numbers by

$$\Park_{W,p}^{J,\pm}(\X)=\prod_{i=1}^{|J|}\frac{[p\pm e_i^{J,p}]_\X}{[d_i^J]_\X},$$

where $d_1^J,\ldots,d_{|J|}^J$ are the fundamental degrees of $W_J$, $e_1^{J,p},\ldots,e_{|J|}^{J,p}$ are the exponents or fake degrees of the $W_J$-action on the Galois-conjugate reflection representation dual $\sf{V}_p^*$, and $[k]_\X=1+\X+\cdots+\X^{k-1}$. The parking-number identity. The crystallographic identity

$$\Park_{W,p}^{J,\pm}(\X)=\frac{1}{(\X-1)^r}\sum_{v\in W^{J,\mp}}\sum_{\vec{\omega}\in\cal{D}^{(v)}(\vec{c}^{p})}\X^{|\sf{d}_{\vec{\omega}}|}(\X-1)^{|\sf{e}_{\vec{\omega}}|}$$

should hold for every irreducible finite Coxeter system, with the same notation and the sign flip as in the crystallographic theorem. This would extend the stated parking-number formula beyond crystallographic Coxeter systems to the full irreducible finite Coxeter setting.