Keller–Lübeck–Lynd–Semikina conjectures for fusion systems

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$ of order $p^e$, and let $\alpha$ be a compatible family. Write $s$ for the sectional rank of $S$, and let $r>0$ be the smallest positive integer such that, when $S$ is nonabelian, $S$ has a character of degree $p^r$. The quantities $\operatorname{\mathbf{k}}(\mathcal{F},\alpha)$, $\operatorname{\mathbf{w}}(\mathcal{F},\alpha)$, and $\operatorname{\mathbf{m}}(\mathcal{F},\alpha,d)$ are defined above; $[S,S]$ denotes the derived subgroup of $S$. Keller–Lübeck–Lynd–Semikina conjectures. The following assertions hold: (1) $\operatorname{\mathbf{k}}(\mathcal{F},\alpha)=\operatorname{\mathbf{m}}(\mathcal{F},\alpha)$; (2) $\operatorname{\mathbf{k}}(\mathcal{F},\alpha)\leqslant |S|$; (3) $\operatorname{\mathbf{w}}(\mathcal{F},\alpha)\leqslant p^s$; (4) for every positive integer $d$, $\operatorname{\mathbf{m}}(\mathcal{F},\alpha,d)\geqslant 0$; (5) if $S$ is nonabelian, then $\operatorname{\mathbf{m}}(\mathcal{F},\alpha,d')\ne 0$ for some $d\ne e$; (6) if $S$ is nonabelian, then $r$ is the smallest positive integer such that $\operatorname{\mathbf{m}}(\mathcal{F},\alpha,d-r)\ne 0$; and (7a) $\operatorname{\mathbf{k}}(\mathcal{F},\alpha)/\operatorname{\mathbf{m}}(\mathcal{F},\alpha,e)$ is at most the number of conjugacy classes of $[S,S]$, while (7b) $\operatorname{\mathbf{k}}(\mathcal{F},\alpha)/\operatorname{\mathbf{w}}(\mathcal{F},\alpha)$ is at most the number of conjugacy classes of $S$. These conjectures generalize weight and ordinary weight conjectures from blocks to compatible families on saturated fusion systems. The paper states that they will be verified for nonconstrained fusion systems on an extraspecial group, while the assertions are presented here as the seven conjectures from the cited source.