Weak Serre conjecture over totally real fields

Let $F$ be a totally real field, let $G_F$ be its absolute Galois group, and let $\overline{\rho}:G_F\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ be a mod $p$ Galois representation. Call $\overline{\rho}$ totally odd when the determinant of its image at every complex conjugation associated with an embedding $F\hookrightarrow\mathbb R$ is $-1$.

Weak Serre conjecture. If $\overline{\rho}$ is continuous, irreducible, and totally odd, then it is modular.

This is the natural totally real generalization of the classical conjecture over $\mathbb Q$. The source states the conjecture but supplies no resolution status in the provided text.