Strong Serre conjecture on modularity and Serre weights

Let $F$ be a totally real field, let $p$ be prime, and let $\overline{\rho}:G_F\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ be a mod $p$ Galois representation. Let $\sigma$ be a Serre weight, and let $W(\overline{\rho})$ denote the set of Serre weights associated with $\overline{\rho}$.

Strong Serre conjecture. If $\overline{\rho}$ is continuous, irreducible, and totally odd, then it is modular. Moreover, $\overline{\rho}$ is modular of weight $\sigma$ if and only if

$$\sigma\in W(\overline{\rho}).$$

The conjecture refines modularity by predicting exactly which Serre weights occur. The source gives no resolution status in the provided text.