The Fontaine–Mazur modularity conjecture over totally real fields

Let $K$ be a totally real field and let $p\geq3$ be prime. A continuous representation

$$\rho:G_K\longrightarrow\operatorname{GL}_2(\overline{\mathbb{Q}}_p)$$

is geometric if it satisfies the geometricity conditions of Fontaine and Mazur. The Fontaine–Mazur modularity conjecture. If $\rho$ is a geometric irreducible continuous Galois representation, then $\rho$ is modular, meaning that it arises from a Hilbert modular form over $K$. This is a broad modularity prediction underlying the modular method. The source cites it as a conjecture and gives no resolution in the stated generality.