Frenkel–Hernandez–Geiss–Hernandez–Leclerc conjecture on Q-variables and simple modules

Let $(i,a)\in I\times_2\mathbb Z$, let $w\in W$, and let $Q_{w\varpi_i^{\vee},a}$ be the corresponding $Q$-variable. Let $\chi_{\ell}$ denote the $\ell$-character and let $L_{w\varpi_i^{\vee},a}$ be a simple object of $\mathcal{O}_{sh}^{\mathbb Z}$. Q-variable realization conjecture. There are explicit simple objects $L_{w\varpi_i^{\vee},a}$ such that

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

In particular, their classes in $K_0(\mathcal{O}_{sh}^{\mathbb Z})$ satisfy the extended $QQ$-system. This conjecture would realize the formal $Q$-variables representation-theoretically and is cited as an earlier proposal; no resolution is stated in the paper.