The weight part of Serre's conjecture in the generic tame setting

Let $F^+$ be the relevant totally real field, let $\overline r:G_{F^+}\rightarrow\mathcal{G}_n(\mathbb{F})$ be automorphic, and let $\overline r_p$ be its associated local $L$-homomorphism at $p$. Let $W(\overline r)$ be the set of modular Serre weights and $W^?(\overline r_p|_{I_{\mathbb{Q}_p}})$ the predicted Serre-weight set attached to the inertial restriction. Weight part of Serre's conjecture. If $\overline r_p|_{I_{\mathbb{Q}_p}}$ is tame and $2n$-generic, then

$$W(\overline r)=W^?(\overline r_p|_{I_{\mathbb{Q}_p}}).$$

This predicts exactly which Serre weights occur for automorphic mod-$p$ representations; the paper presents it as a conjectural formulation based on earlier work of Gee–Herzig–Savitt.