Broué's geometric extension conjecture for the normalizer of a torus

Let $G$ be a finite reductive group, let $\ell$ satisfy the preceding assumptions (B1)–(B3), let $T_w$ be the associated torus, let $b$ be the principal $\ell$-block of $G$, and let $\Lambda$ be the coefficient ring. Broué's geometric extension conjecture. There exists a bounded complex $D$ of $(\Lambda G b,\Lambda N_G(T_w))$-bimodules such that the restrictions of $b\mathrm{R}\Gamma_c(\mathrm{Y}(\dot w),\Lambda)$ and $D$ to bounded complexes of $(\Lambda G b,\Lambda T_w)$-bimodules are homotopy equivalent, and $D$ induces a splendid Rickard equivalence between the principal $\ell$-blocks of $G$ and $N_G(T_w)$. This gives a more precise formulation of the geometric Broué prediction by requiring an equivalence complex extending the Deligne–Lusztig complex from $T_w$ to its normalizer.