Folding conjecture for irreducible $l$-highest weight modules

Let $\mathcal{U}_q(\mathcal{L}\mathfrak{g})$ and $\mathcal{U}_q(\mathcal{L}\mathfrak{g}^{\sigma})$ be the non-twisted and twisted quantum affine algebras, respectively. Let $\mathrm{Rep}_1(\mathcal{U}_q(\mathcal{L}\mathfrak{g}))$ and $\mathrm{Rep}_1(\mathcal{U}_q(\mathcal{L}\mathfrak{g}^{\sigma}))$ be the subrings generated by the indicated $l$-highest weight modules, and let $\bar{\pi}$ be the folding-induced ring isomorphism. For a monomial $M\in\mathbb{Z}[Y_{i,q^r}^{\pm1}]_{i\in I,r\in\mathbb{Z}}$, write $L(M)$ for the corresponding irreducible $l$-highest weight module. Folding conjecture. The isomorphism

$$\bar{\pi}:\mathrm{Rep}_1(\mathcal{U}_q(\mathcal{L}\mathfrak{g}))\xrightarrow{\sim}\mathrm{Rep}_1(\mathcal{U}_q(\mathcal{L}\mathfrak{g}^{\sigma}))$$

maps $[L(M)]$ to $[L(\pi(M))]$ for all such monomials $M$. The claim is known for KR-modules, but the supplied text gives no resolution in the stated generality.