Frenkel–Hernandez conjecture on Langlands dual representations

Let $\mathcal{U}_q(\hat{\mathfrak{g}})$ and $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ be quantum affine algebras of Langlands-dual types, let $P'$ be the sublattice specified in the paper, and let $\Pi:P'\to\leftindex^L{P}$ be the Langlands character map. For an irreducible finite-dimensional representation $V$ of highest weight $\lambda\in P'$, a Langlands dual representation is an irreducible representation ${}^L V$ of $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ with highest weight $\Pi(\lambda)$ and

$$\chi^\sigma({}^L V)\preceq \Pi(\chi(V)),$$

meaning that $\Pi(\chi(V))-\chi^\sigma({}^L V)\in\mathbb{N}[\leftindex^L P]$. Frenkel–Hernandez conjecture. Every such $V$ admits a Langlands dual representation. It is known for Kirillov–Reshetikhin modules, but remains open in general.