Broué's abelian defect group conjecture

Let $k$ be an algebraically closed field of characteristic $p>0$, $G$ a finite group, $B$ a block of the group algebra $kG$ with defect group $D$, and $b$ the Brauer correspondent of $B$ in $kN_G(D)$. Broué's abelian defect group conjecture. If $D$ is abelian, then the block $B$ is derived equivalent to $b$. Broué's conjecture predicts a local-to-global derived equivalence for blocks with abelian defect groups; the supplied status evidence concerns the failure of the statement without the abelianity assumption, not the conjecture as stated.