Broué's regular-element conjecture for splendid Rickard complexes

Assume the regular-element setting of the source: ${\mathbf L}={\mathbf T}=C_{\mathbf G}(D)$ is a torus, ${\mathcal O}Gb$ is the principal block, $\ell\nmid(q-1)$, $F$ acts trivially on the Weyl group, $w_d$ is the specified good regular element of order $d>1$, and $H/C_G(D)\simeq C_W(w_d)$. Let $B_d$ be the braid group of $C_W(w_d)$ and let $Y(w_d)$ be the associated Deligne–Lusztig variety. Broué's regular-element conjecture. There is a complex

$$C\in K^b(({\mathcal O}Gb)\otimes({\mathcal O}Hb_D)^{\operatorname{opp}}\operatorname{-lperm})$$

unique up to isomorphism, together with the surjective morphism and the two compatibility properties stated in the source; moreover, such a complex induces a Rickard equivalence between ${\mathcal O}Gb$ and ${\mathcal O}Hb_D$. This is a precise splendid refinement of the Deligne–Lusztig form of Broué's conjecture. The source records several special cases but leaves the general assertion open.