Geometric Broué–Malle conjecture for Deligne–Lusztig varieties

Let $(K,\mathcal{O},k)$ be an $\ell$-modular system, let $B$ be a unipotent block with defect group $D$, and let $d$ be the order of $q$ modulo $\ell$. Let $C$ be the reduction modulo the maximal ideal of $\mathcal{O}$ of the cohomology complex associated with a Deligne–Lusztig variety $Y_{\kappa/d}$, carrying an action of $C_G(D)$ on the left and $G$ on the right. Broué–Malle conjecture. There exists a positive integer $\kappa$ such that $C$, viewed as a complex of $(kN_G(D),kG)$-bimodules, induces a derived equivalence between $B$ and its Brauer correspondent $b$. This is the geometric refinement of Broué's conjecture through Deligne–Lusztig cohomology; the source presents it as open and notes that only restricted cases are known.