The block-exoticity conjecture for exotic fusion systems

Let $p$ be a prime, let $P$ be a finite $p$-group, and let $\mathcal F$ be a saturated fusion system on $P$. The fusion system $\mathcal F$ is exotic if it is not of the form $\mathcal F_P(G)$ for a finite group $G$ with $P\in\operatorname{Syl}_p(G)$; it is block-exotic if it is not of the form $\mathcal F_{(P,e_P)}(G,b)$ for a finite group $G$ and a $p$-block $b$ with maximal $b$-Brauer pair $(P,e_P)$. Block-exoticity conjecture. If $\mathcal F$ is an exotic fusion system, then $\mathcal F$ is block-exotic. This conjecture asks whether every exotic fusion system fails to arise from a block, whereas every realisable fusion system is block-realisable by the principal-block case of Brauer's Third Main Theorem. It remains open.