Lusztig's conjecture on the function a

Let $W$ be the Coxeter group under consideration, and let ${\mathcal{M}}(w)$ denote the set of maximal elements associated with $w\in W$. Define

$$a'(w)=\max_{v\in {\mathcal{M}}(w)}a(v).$$

Lusztig's conjecture on the function $a$. For every $w\in W$, one has

$$a(w)=a'(w).$$

This is presented as a variant of a conjecture of Lusztig about the function $a$. The supplied passage does not state whether the conjecture has been proved or disproved.