The Galois Alperin weight conjecture for finite category algebras

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 with $p$-orbit category $\mathcal{O}_{\mathcal{C}}$. Let $\mathcal{S}(k\mathcal{C})$ be the set of isomorphism classes of simple $k\mathcal{C}$-modules and let $\mathcal{W}(k\mathcal{O}_{\mathcal{C}})$ be the set of isomorphism classes of weights. The Galois Alperin weight conjecture. 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 $\Gamma$. This is the main Galois refinement proposed for finite category algebras; the abstract further states that it is reduced to the corresponding conjecture for finite groups.