Local mod $p$ Langlands determination conjecture for ordinary representations

Let $F$ be a CM field in which $p$ is unramified, let $w$ be a place of $F$ above $p$, and let $\overline{r}:\operatorname{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ be an automorphic Galois representation. Let $\Pi(\overline{r})$ be the associated smooth representation of $\mathrm{GL}_n(F_w)$, and write $\overline{r}|_{\operatorname{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ for its local restriction. Local mod $p$ Langlands determination conjecture. The local Galois representation $\overline{r}|_{\operatorname{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by $\Pi(\overline{r})$. This conjecture asks whether the automorphic representation and the corresponding local Galois representation determine each other in the mod $p$ Langlands setting; the paper notes that the structure of $\Pi(\overline{r})$ is poorly understood beyond known descriptions of its ordinary part.