Crystalline-lift criterion for weights of mod-$p$ representations

Let $n\geq1$, let $p$ be a prime, and let $\overline\rho:G_{\mathbb Q}\to\operatorname{GL}_n(\overline{\mathbb F}_p)$ be continuous, odd, and irreducible. Let $F(a_1,\ldots,a_n)$ be the simple $\overline{\mathbb F}_p[\operatorname{GL}_n(\mathbb F_p)]$-module of highest weight $(a_1,\ldots,a_n)$, with $0\leq a_i-a_{i+1}\leq p-1$. Crystalline weight conjecture. $F(a_1,\ldots,a_n)$ is a weight for $\overline\rho$ if and only if $\overline\rho|_{G_{\mathbb Q_p}}$ has a crystalline lift with Hodge–Tate weights

$$a_1+(n-1),a_2+(n-2),\ldots,a_n.$$

This gives a conjectural description of the weights of higher-dimensional mod-$p$ Galois representations in terms of local crystalline lifts; the source does not establish the criterion in general.