The gluing conjecture for splendid Rickard equivalences

Let $G$ be a finite group, let $D$ be an abelian Sylow $\ell$-subgroup, let $b$ and $c$ be the principal block idempotents of $kG$ and $kH$ with $H=N_G(D)$, and form the groups and category of $\bar N$-sets described in the source. Let $\tilde{\mathcal E}$ be the Karoubian envelope of the linearization of the category of $\bar N$-sets whose point stabilizers are contained in $\Delta D/Z$. The gluing conjecture. There is a complex $C$ of objects of $\tilde{\mathcal E}$ such that

$$\operatorname{Res}_{G\times H^{\operatorname{opp}}}^{\bar N}k(C)$$

induces a Rickard equivalence between $kGb$ and $kHc$. This formulation incorporates central $\ell$-subgroups and $\ell'$-automorphism groups into the inductive approach to Broué's conjecture. The source gives no resolution.