Folding conjecture for q-characters of quantum affine algebras

Let ${\mathfrak{g}}$ be any finite type Lie algebra with an automorphism $\sigma$. For $i\in I$ and $x\in q^{\mathbb{Z}}$, let $Y_{i,x}$ denote the $I$-tuple of power series that equals

$$\frac{zq_i-xq_i^{-1}}{z-x}$$

on the $i$-th spot and equals $1$ elsewhere, where $q_i=q^{d_i}$ with $d_i\in\{1,2,3\}$. Let $\boldsymbol{\psi}=Y_{i_1,x_1}\dots Y_{i_n,x_n}$ for $i_1,\dots,i_n\in I$ and $x_1,\dots,x_n\in q^{\mathbb{Z}}$.

Folding conjecture. The $q$-characters of the simple modules $L(\boldsymbol{\psi})$ of $U_q(L{\mathfrak{g}})$ and $L(\rho(\boldsymbol{\psi}))$ of $U_q(L{\mathfrak{g}}^\sigma)$ satisfy

$$\rho\bigl(\chi_q(L(\boldsymbol{\psi}))\bigr)=\chi_q\bigl(L(\rho(\boldsymbol{\psi}))\bigr).$$

Here $\rho$ is the folding map from untwisted to twisted loop weights, and the conjecture relates the representation theories of the corresponding untwisted and twisted quantum affine algebras. The statement was conjectured in the cited work and a general version is also attributed to another source; no resolution is supplied here.