Hernandez's conjecture on specialization of $(q,t)$-characters

Let $\mathfrak g$ be a general simple Lie algebra, let $\mathcal M_+$ be the set of dominant monomials, and let $\chi_{q,t}(L(m))$ and $\chi_q(L(m))$ denote the $(q,t)$-character and $q$-character of the simple representation $L(m)$, respectively. Hernandez's conjecture. For any $m\in\mathcal M_+$, the equality

$$\operatorname{ev}_{t=1}\chi_{q,t}(L(m))=\chi_q(L(m))$$

holds. Nakajima proved this when $\mathfrak g$ is of type $\mathrm{ADE}$; the conjecture asks for the same specialization identity for general simple Lie algebras and is stated as open in full generality.