Broué–Malle–Michel specialization conjecture for Deligne–Lusztig endomorphism algebras

Let $\mathbb{G}$ be a generic finite reductive group, let $e>0$, and let $(\mathbb{L},\lambda)$ be a $\Phi_e$-cuspidal pair. Let $H_{W_{\mathbb{G},\mathbb{L},\lambda}}(\vec{u})$ be the generic Hecke algebra attached to the relative complex reflection group, and let $Y_{L\subseteq P}^G$ be the associated Deligne–Lusztig variety. Broué–Malle–Michel's specialization conjecture. There should be a $\Phi_e$-specialization producing an algebra $H_{\mathbb{G},\mathbb{L},\lambda}(x)$ whose specialization at every prime power $q$ is isomorphic to the endomorphism algebra of the relevant Deligne–Lusztig cohomology representation. This conjecture identifies generic Hecke algebras with the endomorphism algebras arising in finite reductive Harish–Chandra theory; the supplied text gives no resolution status.