The inductive Alperin weight condition for finite categories

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, and let $\mathcal{C}$ be a finite category. Let $\operatorname{Aut}(\mathcal{C})$ denote its automorphism group. If $\mathcal{C}$ is an EI-category, let $b$ be a central idempotent of $k\mathcal{C}$ and let $\operatorname{Aut}(\mathcal{C})_b$ denote its stabiliser. Inductive Alperin weight condition. (i) For any finite category $\mathcal{C}$, there exists a bijection

$$\mathcal{S}(k\mathcal{C})\to\mathcal{W}(k\mathcal{O}_{\mathcal{C}})$$

commuting with the action of $\operatorname{Aut}(\mathcal{C})$. (ii) For any finite EI-category $\mathcal{C}$ and any central idempotent $b$ of $k\mathcal{C}$, there exists a bijection

$$\mathcal{S}(k\mathcal{C}b)\to\mathcal{W}(k\mathcal{O}_{\mathcal{C}},b)$$

commuting with the action of $\operatorname{Aut}(\mathcal{C})_b$. This is the category-algebra extension proposed after the finite-group inductive condition; the paper does not report a resolution.