The Knörr–Robinson reformulation for fusion systems

Let $S$ be a nontrivial finite $p$-group, let $\mathcal{F}$ be a saturated fusion system on $S$, and let $\alpha$ be an $\mathcal{F}$-compatible family. For each fully normalized normal chain $\sigma=(Q_0<\cdots<Q_m)$ of nontrivial subgroups, let $N_{\mathcal{F}}(\sigma)$ be the associated saturated fusion system and let $\alpha(\sigma)$ be the induced compatible family. The Knörr–Robinson conjecture. One has

$$\mathbf{k}(\mathcal{F},\alpha)=\sum_{\sigma}(-1)^{|\sigma|}\mathbf{k}\bigl(N_{\mathcal{F}}(\sigma),\alpha(\sigma)\bigr),$$

where the sum is over representatives of the isomorphism classes of fully normalized normal chains of nontrivial subgroups of $S$. This is the fusion-system translation of the Knörr–Robinson reformulation of Alperin's Weight Conjecture; analogous versions with $\mathbf{m}$ or $\mathbf{m}^*$ are mentioned but are not separate conjectures in the supplied candidate span.