Robinson's ordinary weight conjecture for fusion-system block invariants

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $S$, let $\alpha$ be an $\mathcal{F}$-compatible family, and suppose that $(\mathcal{F},\alpha)$ realizes a block $B$ of $kG$ for a finite group $G$. For $d\geqslant 0$, let $\mathbf{m}(\mathcal{F},\alpha,d)$ be the local weight invariant and let $\mathbf{k}_d(B)$ be the number of irreducible characters of defect $d$ in $B$.

Robinson's ordinary weight conjecture. One has

$$\mathbf{m}(\mathcal{F},\alpha,d)=\mathbf{k}_d(B).$$

The statement is presented as the relevance of the local invariant. In the paper's applications, the corresponding equality is proved for the principal $2$-block of $\operatorname{Spin}_7(q)$, while this general formulation is attributed to the ordinary weight conjecture framework.