Alperin's weight conjecture for blocks

Let ${\mathcal{O}}Gb$ be a block of a finite group $G$ with defect group $D\neq 1$. For a chain ${\mathcal{S}}$ of subgroups

$$1<Q_1<Q_2<\cdots<Q_n\leq_G D,$$

let $l({\mathcal{S}})=n\geq 1$, let the sum run over conjugacy classes of such chains, and let $c_{\mathcal{S}}$ be the sum of the block idempotents of $N_G({\mathcal{S}})$ corresponding to $b$. Alperin's conjecture.

$$\operatorname{rank} K_0(kGb)=\sum_{\mathcal{S}}(-1)^{l({\mathcal{S}})+1}\operatorname{rank}K_0(kN_G({\mathcal{S}})c_{\mathcal{S}}).$$

The assertion expresses the number of simple modules in a block through local subgroup data. It is a central local-global conjecture in modular representation theory, and no resolution is supplied here.