At present, the mod $p$ (and $p$-adic) local Langlands correspondence is only well understood for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. One of the main difficulties is that little is known about supersingular representations besides this case, and we do know that there is no simple one-to-one correspondence between representations of $\mathrm{GL}_2(K)$ with two-dimensional representations of $\mathrm{Gal}(\overline{K}/K)$, at least when $K/\mathbb{\mathbb{Q}}_p$ is (non-trivial) finite unramified.
However, the Buzzard-Diamond-Jarvis conjecture and the mod $p$ local-global compatibility for $\mathrm{GL}_2/\mathbb{Q}$ suggest that this hypothetical correspondence may be realized in the cohomology of Shimura curves with characteristic $p$ coefficients (cut out by some modular residual global representation $\bar{r}$). Moreover, the work of Gee, Breuil and Emerton-Gee-Savitt show that, to get information about the $\mathrm{GL}_2(K)$-action on the cohomology, one could instead study the geometry of certain Galois deformation rings of the $p$-component of $\bar{r}$.
In a work in progress with Haoran Wang, we push forward their analysis of the structure of potentially Barsotti-Tate deformation rings and, as an application, we prove a multiplicity one result of the cohomology at full congruence level when $\bar{r}$ is reducible generic \emph{non-split} at $p$. (The semi-simple case was previously proved by Le-Morra-Schraen and by ourselves.)
