Oliver's p-group conjecture

Let $p$ be an odd prime and let $S$ be a finite $p$-group. The Thompson subgroup $J(S)$ is generated by all elementary abelian $p$-subgroups of $S$ whose rank equals the $p$-rank of $S$, and let $\mathfrak{X}(S)$ be the Oliver subgroup.

Oliver's p-group conjecture.

$$J(S)\leq \mathfrak{X}(S).$$

Oliver's conjecture is a group-theoretic problem with implications for the existence and uniqueness of centric linking systems for fusion systems at odd primes, and hence for the Martino–Priddy conjecture. The Martino–Priddy conjecture and the existence and uniqueness of centric linking systems have been solved, but Oliver's p-group conjecture remains open in general; this paper proves it for Sylow $p$-subgroups of unitary groups.