Linckelmann's nonvanishing conjecture for Hochschild cohomology of blocks

Let $G$ be a finite group, let $k$ be a field whose characteristic divides the order of $G$, and let $B$ be a block of the group algebra $kG$ with defect group $P$. Linckelmann's conjecture. If $P$ is nontrivial, then

$$\mathrm{HH}^1(B,B)\neq 0.$$

The claim is a block-wise analogue of the nonvanishing of outer derivations for finite group algebras. The paper describes it as a conjecture posed by Linckelmann; its status is not resolved in the supplied context.