The Fontaine–Mazur–Langlands automorphy conjecture for conjugate self-dual representations

Let $K/K^+$ be a CM extension. Let $R:G_K\to\operatorname{GL}_m(\overline{\mathbf Q}_p)$ be a continuous irreducible potentially semistable representation, and let $\Psi:G_{K^+}\to\overline{\mathbf Q}_p^*$ be a continuous character such that

$$R^{\vee}\cong R^c\otimes\Psi|_{G_K}.$$

Fontaine–Mazur–Langlands conjecture. There exists an algebraic essentially conjugate self-dual cuspidal automorphic representation $\pi$ of $\operatorname{GL}_m/K$ such that $\rho_\pi\cong R$. This combines the Fontaine–Mazur conjecture with Langlands' automorphy conjectures; the source reports results toward it, but the general statement remains unresolved.