Kottwitz's involution-intersection conjecture for left cells

Let $W$ be a finite Coxeter group, let $C$ be a conjugacy class of involutions in $W$, and let $V_C$ be the real vector space with basis $\{a_w\mid w\in C\}$ equipped with the $W$-action described in the source. Let $\Gamma$ be a left cell of $W$, and let $[\Gamma]_1$ denote its associated representation. Kottwitz's conjecture.

$$\dim \operatorname{Hom}_W(V_C,[\Gamma]_1)=|C\cap\Gamma|.$$

This conjecture relates multiplicities in the involution module $V_C$ to the number of involutions from $C$ contained in a left cell. The source introduces it as Kottwitz's conjecture but gives no resolution in the supplied text.