Serre's weak conjecture over totally real fields

Let $L$ be a totally real number field, let $\ell$ be an odd prime, and let

$$\overline{\rho}:G_L\to\operatorname{GL}_2(\overline{\mathbb{F}}_\ell)$$

be an irreducible and totally odd Galois representation, where totally odd means that $\det(\overline{\rho}(c))=-1$ for every complex conjugation $c$. Serre's weak conjecture. There exists a Hilbert modular eigenform $f$ for $L$ such that $\overline{\rho}$ is isomorphic to the reduction modulo a prime $\lambda$ dividing $\ell$ of the $\lambda$-adic representation $\rho_{f,\lambda}$ attached to $f$. The paper assumes this statement for its results when the auxiliary field is nontrivial; the supplied text gives no resolution evidence.