Weight elimination conjecture for ordinary mod $p$ Galois representations

Let $\overline{r}:G_F\rightarrow\mathrm{GL}_n(\mathbf{F})$ be a continuous automorphic Galois representation with $\overline{r}|_{G_{F_w}}\cong\overline{\rho}_0$, where $\overline{\rho}_0$ is the ordinary representation specified in the source. Let $W_w(\overline{r})$ be the set of local Serre weights, let $\mathrm{JH}(-)$ denote the Jordan--Hölder constituents, and let $\pi^{i_1,j_1}$, $\mu^{\square}$, and $\mu^{\square,i_1,j_1}$ be the principal series and weights defined in the source. Fix integers $(i_0,j_0)$ with $0\leq j_0<j_0+1<i_0\leq n-1$. Weight elimination conjecture. If $\overline{\rho}_{i_0,j_0}$ is Fontaine--Laffaille generic and $\mu^{\square,i_1,j_1}$ is $2n$-generic, then

$$W_w(\overline{r})\cap\mathrm{JH}\bigl((\pi^{i_1,j_1})^{\vee}\bigr)\subseteq\bigl\{F(\mu^{\square})^{\vee},F(\mu^{\square,i_1,j_1})^{\vee}\bigr\}.$$

This is a proposed necessary result for the paper's main local-global compatibility theorem: it restricts which Serre weights can occur under the stated genericity hypotheses. The source presents it as a main conjecture for weight elimination, and no resolution is supplied.