Buzzard–Diamond–Jarvis conjecture for the cohomology of Shimura curves

Let $F$ be a totally real field, let $D$ be an indefinite quaternion algebra over $F$, and let $S^D(\mathbb F)$ be the direct limit of the mod $p$ étale cohomology of the associated Shimura curves. Let $\overline r:G_F\to\mathrm{GL}_2(\mathbb F)$ be modular, and for each finite place $w$ let $\overline\rho_w=\overline r|_{G_{F_w}}$. Let $\pi^D_w(\overline r)$ be the associated smooth admissible representations, characterized at $w\mid p$ by the Serre-weight condition. Buzzard–Diamond–Jarvis' conjecture. There is a $G_F\times(D\otimes_F\mathbb A_{F,f})^\times$-equivariant isomorphism

$$S^D(\mathbb F)[\mathfrak m_{\overline r}]\cong\overline r\otimes\bigl(\otimes'_w\pi^D_w(\overline r)\bigr).$$

At $w\mid p$, $\operatorname{Hom}_{\mathrm{GL}_2(\mathcal O_{F_w})}(\sigma,\pi^D_w(\overline{\rho}))\neq0$ if and only if $\sigma\in\mathscr D(\overline{\rho}_w)$. The weight part is proved by Gee, Liu and Savitt under the usual Taylor–Wiles hypothesis, and multiplicity one is proved by Emerton, Gee and Savitt.