The non-exoticity conjecture for generalised block fusion systems

Let $p$ be a prime, let $H$ be a finite group, let $G\unlhd H$, and let $b$ be an $H$-stable $p$-block of $kG$, where $k$ is an algebraically closed field of characteristic $p$. Let $(P,e_P)$ be a maximal $(b,H)$-Brauer pair. The associated generalised block fusion system is the category $\mathcal F_{(P,e_P)}(H,G,b)$ on $P$ whose morphisms are the injective homomorphisms induced by conjugation by elements of $H$ that preserve the relevant Brauer pairs.

Non-exoticity conjecture. The generalised block fusion system $\mathcal F_{(P,e_P)}(H,G,b)$ is non-exotic.

Generalised block fusion systems are introduced to handle descent to normal subgroups, where ordinary block fusion systems need not behave well under Clifford theory. This conjecture predicts that these systems are always realised by finite-group fusion, and its resolution is not supplied in the source.