The blockwise Galois Alperin weight conjecture for finite EI-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 EI-category, meaning that every endomorphism is an isomorphism. Let $b$ be a central idempotent of $k\mathcal{C}$, let $\Gamma_b$ be its stabiliser in $\Gamma$, let $\mathcal{S}(k\mathcal{C}b)$ be the set of isomorphism classes of simple $k\mathcal{C}b$-modules, and let $\mathcal{W}(k\mathcal{O}_{\mathcal{C}},b)$ be the set of $b$-weights. Blockwise Galois Alperin weight conjecture. 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 $\Gamma_b$. The paper explains that the general category version requires the EI hypothesis to define the relevant partition of weights by blocks, and reduces this conjecture to finite groups.