Breuil–Paškūnas conjecture on the structure of the mod $p$ Langlands representation

Let $\overline{\rho}=\overline{r}_v^{\vee}$ be generic in the sense of Breuil–Paškūnas, and let $\pi_v^D(\overline{r})$ denote the associated mod $p$ Langlands representation. Write $f$ for the integer appearing in the reducible case, and let $\overline{r}^{\rm ss}$ denote the semisimplification of $\overline{r}$. Breuil–Paškūnas conjecture. The representation $\pi_v^D(\overline{r})$ has finite length. More precisely, if $\overline{\rho}$ is irreducible, then $\pi_v^D(\overline{r})$ is irreducible; if $\overline{\rho}$ is reducible, then $\pi_v^D(\overline{r})$ has length $f$ and admits a unique Jordan–Hölder filtration

$$\pi_0\ \textbf{---}\ \pi_1\ \textbf{---}\ \cdots\ \textbf{---}\ \pi_{f-1}\ \textbf{---}\ \pi_f.$$

Here $\pi_0$ and $\pi_f$ are principal series explicitly determined by $\overline{\rho}$, while $\pi_i$ is supersingular for $1\leq i\leq f-1$. Moreover,

$$\pi_v^D(\overline{r}^{\rm ss})=\pi_v^D(\overline{r})^{\rm ss}.$$

This conjecture describes the expected precise structure of the mod $p$ Langlands correspondence in the generic case; the supplied source does not state whether it has been resolved.