Generalized Gessel's conjecture for two-sided Eulerian polynomials

Let $W$ be a finite Coxeter group of rank $n$, and let

$$W(x,y)=\sum_{w\in W}x^{\operatorname{des}_L(w)}y^{\operatorname{des}_R(w)}$$

be its two-sided Eulerian polynomial. Generalized Gessel's conjecture. There exist nonnegative integers $\gamma_{a,b}^W$ such that

$$W(x,y)=\sum_{0\leq 2a+b\leq n}\gamma_{a,b}^W(xy)^a(x+y)^b(1+xy)^{n-2a-b}.$$

This generalizes Gessel's conjecture from the symmetric group to all finite Coxeter groups. The claimed positivity concerns expansion in the basis determined by the symmetries $W(x,y)=W(y,x)$ and $W(x,y)=x^ny^nW(1/x,1/y)$; the source does not state whether the conjecture has been resolved.