Q-character formula for shifted quantum affine simple representations

Let $I$ be the set of simple-root indices, let $W$ be the Weyl group, and let $a\in\mathbb{C}^*$. For $i\in I$ and $w\in W$, let $Q_{w(\varpi_i),a}$ be the corresponding $Q$-variable, and let $\Psi_{w(\varpi_i),a}$ be its leading monomial. Write $L(\Psi_{w(\varpi_i),a})$ for the associated simple representation and $\chi_q$ for its $q$-character. Q-character formula conjecture. One has

$$\chi_q(L(\Psi_{w(\varpi_i),a}))=Q_{w(\varpi_i),a}.$$

The formula would identify the explicitly constructed $Q$-variables with $q$-characters of simple representations. It was formulated in the cited work of Frenkel and Hernandez; the supplied text gives no resolution status.