Geometric Broué's abelian defect conjecture

Let $\ell$ be a prime and let $G=\mathbb{G}(q)$ be a finite group of Lie type, where $p\ne\ell$ and $q$ is a power of $p$. Let $D\in\operatorname{Syl}_{\ell}(G)$, let $(K,\mathcal{O},k)$ be an $\ell$-modular system, and put $H=N_G(D)$. Let $d$ be the order of $q$ modulo $\ell$, and let $\kappa$ be coprime to $d$. Geometric Broué's abelian defect conjecture. There exists a Deligne–Lusztig variety $Y_{\kappa/d}$, with $G$ acting on the right and $C_G(D)$ acting on the left, such that the action of $C_G(D)$ extends to $H$, every unipotent character $\chi\in\operatorname{Uch}(B_0(G))$ occurs in a unique cohomology group $H^i(Y_{\kappa/d},K)$, the complex $C=R\Gamma(Y_{\kappa/d},k)$ induces a perverse equivalence between $B_0(G)$ and $B_0(H)$, and the associated perversity function is obtained from the cohomological degree map via a suitable bijection between $\operatorname{Uch}(B_0(G))$ and $\mathcal{S}_{B_0(H)}$. The source presents this combined geometric statement as open, with only restricted cases known.