The Picard group conjecture for blocks with TI defect groups

Let $B$ be a block of a finite group $G$. The subgroup $\mathcal{E}(B)$ of $\operatorname{Pic}(B)$ consists of the Morita auto-equivalences represented by bimodules with endo-permutation source as $\mathcal{O}(G\times G)$-modules.

Picard group conjecture.

$$\operatorname{Pic}(B)=\mathcal{E}(B).$$

The subgroup $\mathcal{E}(B)$ is an important Morita invariant, and it is not known in general whether it is proper in the Picard group. The conjecture asserts that every Morita auto-equivalence of $B$ has endo-permutation source.