Higgsing conjecture for Kirillov–Reshetikhin $qq$-characters

Let $\Gamma$ be a finite-type quiver and fix $i\in\Gamma_0$. Consider a weight dimension vector with a single nonzero entry, $w_i\neq 0$ and $w_j=0$ for $j\neq i$, and let $\underline{x}=(x_{i,\alpha})$ be the weight parameters. Higgsing conjecture for Kirillov–Reshetikhin modules. The following assertions hold: (1) the $qq$-character is reduced to the $qq$-character of the irreducible highest-weight module if the weight parameters obey the $q_1$-segment condition; (2) this irreducible $qq$-character is further reduced to the irreducible $q_1$-character in the limit $q_2\to 1$; and (3) in the other limit, $q_1\to 1$, the irreducible $qq$-character factorizes into the product of the $q_2$-characters, and hence is not irreducible in general. This is the paper’s more specific conjecture for the Kirillov–Reshetikhin module of a finite-type quiver; the stated factorization also describes the failure of irreducibility in the $q_1\to 1$ limit, while no proof or resolution is supplied.