The quantum Grothendieck ring evaluation conjecture

Let $\mathcal{Y}$ be the monomial ring, let $m$ be a dominant monomial in $\mathcal{Y}$, let $L_t(\underline{m})$ denote the $(q,t)$-character of the simple module $L(m)$, and let $\chi_q(L(m))$ denote its $q$-character. Quantum Grothendieck ring evaluation conjecture. For all dominant monomials $m$ in $\mathcal{Y}$, we have

$$\mathrm{ev}_{t=1}(L_t(\underline{m}))=\chi_q(L(m)).$$

This extends the equality proved by Nakajima in the simply-laced case and is conjectured to hold in general; the statement remains open beyond the cases covered by the known geometric proof.