Frenkel–Hernandez positivity conjecture for affine Langlands branching

Let $\mathcal{U}_q(\hat{\mathfrak{g}})$ and $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ be Langlands-dual quantum affine algebras, let $P'$ be the specified sublattice of the weight lattice, and let $\Pi$ send characters of representations whose weights lie in $P'$ to characters of the Langlands-dual algebra. Frenkel–Hernandez affine positivity conjecture. For every irreducible finite-dimensional representation $V$ of $\mathcal{U}_q(\hat{\mathfrak{g}})$ whose highest weight lies in $P'$, there is a representation $W$ of $\mathcal{U}_q(\leftindex^L{\hat{\mathfrak{g}}})$ such that

$$\chi^\sigma(W)=\Pi(\chi(V)).$$

Equivalently, $\Pi(\chi(V))$ is a positive sum of characters of irreducible representations of the Langlands-dual algebra. This is the stronger affine positivity statement and is presented as conjectural.