Broué–Malle conjecture on Hecke algebras for relative Weyl groups

Let $G$ be a finite reductive group, and let $(M,\mu)$ be a $\Phi_m$-cuspidal pair with relative Weyl group $W_{M,\mu}^G$. Let $H_{M,\mu}^G(x)$ be the Broué–Malle algebra, and let $\bar{\mathbf{Q}}_\ell$ be specialized by $x\mapsto q$. Broué–Malle conjecture. The isomorphism

$$\bar{\mathbf{Q}}_\ell \otimes_{\bar{\mathbf{Q}}[x^{\pm 1/\infty}]} H_{M,\mu}^G(x) \simeq \operatorname{End}_G(\mathrm{H}_c^\ast(Y_\mathbf{P}^\mathbf{G},\bar{\mathbf{Q}}_\ell))$$

holds for all $G,m,M,\mu$. This would extend Lusztig's comparison between specialized Hecke algebras and endomorphism algebras of cohomology to the general Broué–Malle setting; the source reports numerical consistency but does not establish the assertion.