The commutator and conjugacy-class bounds for fusion-system invariants

Let $S$ be a finite $p$-group of order $p^d$, let $\mathcal{F}$ be a saturated fusion system on $S$, and let $\alpha$ be an $\mathcal{F}$-compatible family. Let $\mathbf{k}(\mathcal{F},\alpha)$, $\mathbf{m}(\mathcal{F},\alpha,d)$, and $\mathbf{w}(\mathcal{F},\alpha)$ be the invariants defined in the paper. The commutator and conjugacy-class bounds. Both inequalities hold:

1. $\mathbf{k}(\mathcal{F},\alpha)/\mathbf{m}(\mathcal{F},\alpha,d)$ is at most the number of conjugacy classes of $[S,S]$.
2. $\mathbf{k}(\mathcal{F},\alpha)/\mathbf{w}(\mathcal{F},\alpha)$ is at most the number of conjugacy classes of $S$. These bounds refine the preceding numerical conjectures by comparing local invariants with the conjugacy structure of $S$ and its commutator subgroup. The source does not provide a resolution for the arbitrary fusion-system assertion.