The combined Galois-automorphism Alperin weight conjecture for finite categories

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, and let $\mathcal{C}$ be a finite category. If $\mathcal{C}$ is a finite EI-category, let $b$ be a central idempotent of $k\mathcal{C}$; otherwise let $b=1$. Let $\mathcal{O}_{\mathcal{C}}$ be the $p$-orbit category, let $\mathcal{S}(k\mathcal{C}b)$ be the set of isomorphism classes of simple modules, and let $\mathcal{W}(k\mathcal{O}_{\mathcal{C}},b)$ be the set of $b$-weights. Write $(\Gamma\times\operatorname{Aut}(\mathcal{C}))_b$ for the stabiliser of $b$. Combined Galois-automorphism conjecture. Let $\mathcal{C}$ be a finite category (respectively, EI-category) and let $b$ be the unit element (respectively, a central idempotent) of $k\mathcal{C}$. Then there exists a bijection

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

commuting with the action of $(\Gamma\times\operatorname{Aut}(\mathcal{C}))_b$, the stabiliser of $b$ in $\Gamma\times\operatorname{Aut}(\mathcal{C})$. This combines the Galois and automorphism refinements and is presented as a stronger conjecture; no resolution is given in the supplied text.