Aschbacher–Kessar–Oliver's exotic fusion systems conjecture

Let $p$ be a prime and let $S$ be a finite $p$-group. A fusion system on $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective group homomorphisms satisfying the fusion-system axioms. It is exotic if it is not realizable as the fusion system of a finite group, and block-exotic if it is not induced by a block of a group algebra.

Aschbacher–Kessar–Oliver's conjecture. If a fusion system $\mathcal{F}$ on $S$ is exotic, then $\mathcal{F}$ is block-exotic.

Every realizable fusion system is block-realizable, because the principal block induces the group fusion system. Thus the conjecture asks whether the converse implication also holds. Several families of fusion systems support it, but the general problem remains open.