Generalization of the centralizer theorem for parabolic subgroups in Coxeter groups

Let $(W,S)$ be a Coxeter system and let $I\subseteq S$. Write $W_I$ for the standard parabolic subgroup generated by $I$, let $\Pi^I$ denote the corresponding subspace of the geometric representation, let $Y_I$ be the subgroup of $Z_W(W_I)$ introduced in the paper, and let $W^{\perp I}{}_{\mathrm{fin}}$ be the finite-type part of the perpendicular subgroup $W^{\perp I}$. For $\gamma\in\Pi^I$, write $s_\gamma$ for the reflection associated with $\gamma$.

Generalized centralizer theorem. For arbitrary $I\subseteq S$, if $\gamma\in\Pi^I$ and $s_\gamma\in W^{\perp I}{}_{\mathrm{fin}}$, then $w\cdot\gamma=\gamma$ for every $w\in Y_I$. Consequently, every element of $Y_I\subseteq Z_W(W_I)$ commutes with every element of $W^{\perp I}{}_{\mathrm{fin}}$.

The claim removes the theorem's restriction that $I$ have no irreducible components of type $A_n$ for $2\leq n<\infty$, and would extend the description of centralizers of standard parabolic subgroups in general Coxeter groups. The source presents it as an assertion the author believes but whose proof has not been given; its resolution is not established here.