The Broué–Malle–Michel cohomological conjecture for good elements

Let $\mathbf G$ be an algebraic group over $\bar{\mathbb F}_q$, let $F$ be a Frobenius map, let $G=\mathbf G^F$, and let $W$ be the $F$-split Weyl group. Let $w\in W$ be a good element of order $d$, let $X_w$ be the associated Deligne–Lusztig variety, and fix a prime $\ell$ not dividing $q$. Write $H^i_c(X_w,\bar{\mathbb Q}_\ell)$ for its étale cohomology, and let $\mathcal H(\mathbf G,F,W,w)$ be the image of $\bar{\mathbb Q}_\ell B_w$ in the graded endomorphism algebra of $\bigoplus_{i\geq0}H^i_c(X_w,\bar{\mathbb Q}_\ell)$. Broué–Malle–Michel's conjecture. Suppose that $w$ is a good element of order $d$. Then: (1) if $i\ne j$, the $\bar{\mathbb Q}_\ell\mathbf G^F$-modules $H^i_c(X_w,\bar{\mathbb Q}_\ell)$ and $H^j_c(X_w,\bar{\mathbb Q}_\ell)$ have no irreducible constituents in common; (2) there is a $d$-cyclotomic Hecke algebra $\mathscr H_x(C_W(w))$ such that

$$\mathcal H(\mathbf G,F,W,w)\cong\mathscr H_{\bar{\mathbb Q}_\ell,q}(C_W(w))\cong\operatorname{End}_{\bar{\mathbb Q}_\ell\mathbf G^F}\left(\bigoplus_{i\geq0}H^i_c(X_w,\bar{\mathbb Q}_\ell)\right);$$

(3) there is a one-to-one correspondence $\chi\longleftrightarrow\chi_q$ between irreducible representations of $C_W(w)$ and irreducible constituents of $\bigoplus_{i\geq0}H^i_c(X_w,\bar{\mathbb Q}_\ell)$, and for each irreducible character $\chi$ there is a polynomial $D_\chi(x)$, depending only on $\chi$, such that $\deg(\chi_q)=D_\chi(q)$. The conjecture proposes a uniform description of the cohomology and its endomorphism algebra in terms of cyclotomic Hecke algebras; the supplied text does not state whether it has been resolved.