The double coset conjecture for finite Coxeter groups

Let $W$ be a finite Coxeter group with simple generators $S$. For $I,J\subset S$, let $X_{IJ}$ be the set of distinguished representatives used in the double-coset decomposition, and for $b\in X_{IJ}$ define

$$W(I,J,b)=\{w\in W\mid w^Jb^{-1}=(wb^{-1})_I\},$$

where $(w^J,w_J)$ and $((wb^{-1})^I,(wb^{-1})_I)$ are the parabolic components of $w$ and $wb^{-1}$, respectively. Double coset conjecture. If $I,J\subset S$, then for all $b\in X_{IJ}$,

$$|W(I,J,b)|=|W(J,I,b^{-1})|.$$

Hohlweg and collaborators proved the equality when $b=e$, while the conjecture concerns arbitrary $b\in X_{IJ}$ and is not established in the supplied text.