Order-polynomial construction of commutative descent algebras for finite Coxeter groups

Let $W$ be a finite Coxeter group, let $w\in W$, and let $\Omega_W(w;t)$ denote the order polynomial associated with $W$ and $w$. Assume that $t_W$ is a linear function depending only on $W$. Order-polynomial commutativity conjecture. For all $x,y$, one has

$$\Omega_W(w;t_W(xy))=\sum_{uv=w}\Omega_W(u;t_W(x))\Omega_W(v;t_W(y)).$$

This identity would provide, via order polynomials, a commutative subalgebra of $\mathbb{Q}[W]$ for every finite Coxeter group, extending the type A and type B constructions. The source presents it as a conjecture; no resolution is supplied here.