Conjecture on folding identities for arbitrary dominant monomials

Let $\mathfrak g$ be simply laced, let $\sigma$ be the folding data, and let $\phi^\sigma_{\mathfrak g}$ and $\Phi^\sigma_{\mathfrak g}$ be the monomial and Grothendieck-ring folding homomorphisms, with

$$\phi^\sigma_{\mathfrak g}\bigl(\chi_q(L(m))\bigr)=\chi_q^\sigma\bigl(L^\sigma(\phi^\sigma_{\mathfrak g}(m))\bigr)$$

for Kirillov--Reshetikhin modules. Folding identity conjecture. The equalities

$$\Phi^\sigma_{\mathfrak g}[L(m)]=[L^\sigma(\phi^\sigma_{\mathfrak g}(m))]$$

and

$$\phi^\sigma_{\mathfrak g}\bigl(\chi_q(L(m))\bigr)=\chi_q^\sigma\bigl(L^\sigma(\phi^\sigma_{\mathfrak g}(m))\bigr)$$

hold for any dominant monomial $m$. Hernandez's theorem establishes these equalities for Kirillov--Reshetikhin modules; the conjecture extends them to all dominant monomials.