Galois Alperin weight conjecture

Fix a prime $p$ and a finite group $G$. Let $\mathcal{H}=\operatorname{Gal}(\mathcal{K}/\mathbb{Q}_p)$ be the finite abelian group of Galois automorphisms associated with the $p$-modular system described in the source. Let $\operatorname{IBr}(G)$ be the irreducible Brauer characters of $G$, and let $\mathcal{W}(G)$ be the set of $p$-weights of $G$. The group $\mathcal{H}\times\operatorname{Aut}(G)$ acts on both sets, and hence on the set of $G$-orbits $\mathcal{W}(G)/\sim_G$. Galois Alperin weight conjecture. For any finite group $G$, there exists an $\mathcal{H}\times\operatorname{Aut}(G)$-equivariant bijection

$$\operatorname{IBr}(G)\longrightarrow\mathcal{W}(G)/\sim_G.$$

This strengthens Alperin's weight conjecture by requiring compatibility with Galois and group automorphisms. It is known for $p$-solvable groups in the strengthened form, but remains open in general.