Navarro's Galois Alperin weight conjecture for finite groups

Let $p$ be a prime, let $k$ be an algebraic closure of $\mathbb{F}_p$, let $\Gamma={\rm Gal}(k/\mathbb{F}_p)$, let $G$ be a finite group, and let $\mathcal{O}_G$ be its $p$-orbit category. Let $\mathcal{S}(kG)$ and $\mathcal{W}(k\mathcal{O}_G)$ denote the sets of isomorphism classes of simple modules and weights, respectively. Navarro's conjecture. For any finite group $G$, there exists a bijection

$$\mathcal{S}(kG)\to\mathcal{W}(k\mathcal{O}_G)$$

commuting with the action of $\Gamma$. This is the Galois refinement of Alperin's weight conjecture; the paper uses it as the finite-group case of its category-algebra formulation.