Realization of block fusion systems as group fusion systems

Determine whether, for any prime p, the fusion systems arising from p-blocks of finite groups are exactly those that can be realized as p-fusion systems F_S(G) of finite groups.

Background

The paper explains that every p-block of a finite group yields a saturated fusion system over its defect group, with the principal block yielding the usual p-fusion system of the group. The authors note a widely discussed conjecture connecting these block-derived fusion systems with the broader class of group-based fusion systems.

Establishing this equivalence would clarify the relationship between block theory in modular representation and the categorical structure of fusion systems, and would help to delineate the scope of exotic fusion systems in representation-theoretic contexts.