Frenkel–Hernandez's Q-variable realization conjecture

Let $(i,a)\in I\times_2\mathbb Z$ and $w\in W$, and let $Q_{w\varpi_i,a}$ be the corresponding formal solution of the extended $QQ$-system. Let $\chi_{\ell}$ denote the $\ell$-character. Q-variable realization conjecture. There is a simple module $L_{w\varpi_i,a}$ such that

$$Q_{w\varpi_i,a}=\chi_{\ell}(L_{w\varpi_i,a}).$$

This is the representation-theoretic realization of the formal $Q$-variables and is an earlier conjecture recalled in the paper; its status is not resolved in the supplied text.