Block-theoretic analogue of Glauberman's Z* theorem

Let $G$ be a group, let $e$ be a block of the group algebra $\mathcal O G$, and let $(P,e_P)$ be an $e$-subpair of $G$. Let $(D,e_D)$ be a maximal $e$-subpair containing $(P,e_P)$, and suppose that $C_G(P)$ controls the $e$-fusion in $G$ with respect to $(D,e_D)$. Block-theoretic Z conjecture.* There exists a Morita equivalence between the block algebras

$$\mathcal O G e \quad\text{and}\quad \mathcal O C_G(P)e_P.$$

This conjecture is a block-theoretic analogue of Glauberman's $Z^*$-theorem. It would strengthen the stable equivalence obtained from strong fusion control to a Morita equivalence, but the supplied source does not state whether it has been resolved.