Diamond–Sasaki geometric modularity conjecture

Let $F$ be a totally real field in which $p$ is unramified, let $G_F$ be its absolute Galois group, and let $\Sigma_v$ denote the embeddings associated with a place $v\mid p$. Let $\Xi_{\min}^{+}$ be the set of weights specified by Diamond and Sasaki. Suppose that $\rho:G_F\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ is geometrically modular of some weight. Diamond–Sasaki's geometric modularity conjecture. If $k\in\Xi_{\min}^{+}$, then $\rho$ is geometrically modular of weight $(k,l)$ if and only if, for every $v\mid p$, the restriction $\rho|_{G_{F_v}}$ has a crystalline lift of weight $(k_{\tau},l_{\tau})_{\tau\in\Sigma_v}$. This remains open.