Irreducibility conjecture for the folding map on quantum affine representations

Let $\mathfrak{g}$ be of type $A_{2n-1}$, let $\bar\pi$ be the induced homomorphism from the Grothendieck ring of finite-dimensional representations of the non-twisted quantum affine algebra $\mathcal{U}_q(\hat{\mathfrak{g}})$ to that of the twisted algebra $\mathcal{U}_q(\hat{\mathfrak{g}}^\sigma)$, and restrict to representations whose $l$-weights are monomials in

$$\mathbb{Z}[Y_{i,q^n}^{\pm1}]_{i\in I,n\in\mathbb{Z}}.$$

For a dominant monomial $m$ in this ring, let $\pi(m)$ be its corresponding dominant monomial in the twisted monomial algebra $\mathcal{Z}$. Folding irreducibility conjecture. The folding map satisfies

$$\bar\pi([L(m)])=[L(\pi(m))].$$

Thus it maps the class of every irreducible representation in the restricted subcategory to the class of an irreducible representation. The statement is given as conjectural; no resolution is supplied in the paper.