Buzzard–Diamond–Jarvis conjecture for quaternionic mod-$p$ cohomology

Let $F$) be a totally real field and let

$$\rho: G_F:=\operatorname{Gal}(\mathbb{\bar{Q}}/F)\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$$

be a continuous, irreducible and totally odd representation. Let $D$ be a quaternion algebra over $F$ that splits at exactly one infinite place, and let $S^D[\mathfrak{m}_\rho]$ be the corresponding localized mod-$p$ cohomology representation of $G_F\times D_f^\times$. Buzzard–Diamond–Jarvis conjecture. The representation $S^D[\mathfrak{m}_\rho]$ is isomorphic to a restricted tensor product

$$S^D[\mathfrak{m}_\rho]\cong \rho\otimes (\otimes'_w\pi_w),$$

where each $\pi_w$ is a smooth admissible representation of $D_w^\times$, with the following properties: if $w$ does not divide $p$, then $\pi_w$ is the representation attached to $\rho_w:=\rho|_{F_w}$ by the modulo $\ell$ local Langlands or Jacquet–Langlands correspondence; if $w$ divides $p$, then $\pi_w\ne 0$, and, when both $F$ and $D$ are unramified at $w$, for every irreducible representation $\sigma$ of $\operatorname{GL}_2(\mathcal{O}_{F_w})$,

$$\operatorname{Hom}_{\operatorname{GL}_2(\mathcal{O}_{F_w})}(\sigma,\pi_w)\ne 0$$

if and only if $\sigma\in W(\rho_w)$, where $W(\rho_w)$ is the associated set of Serre weights. This conjecture predicts a precise local–global description of mod-$p$ automorphic forms and relates the local factors at places above $p$ to Serre weights; the source does not state a resolution.