Hernandez–Leclerc's quiver-moduli formula for q-characters

Let ${\mathfrak{g}}$ be of finite type and let ${\boldsymbol{\psi}}$ be a reachable highest $\ell$-weight. Write ${\boldsymbol{\psi}}$ in terms of vectors ${\mathbf{l}}=(l_{i,s})$ and ${\boldsymbol{m}}=(m_{i,s})$ of non-negative integers, and let $N^{\emph{stab}}_{\boldsymbol{x},{\boldsymbol{m}},{\mathbf{l}}}$ be the closed subvariety of the stable graded-quiver representation moduli space defined by the critical-locus and vanishing conditions in the statement. Hernandez–Leclerc's conjecture. The $q$-character satisfies

$$\chi_q(L({\boldsymbol{\psi}}))=[{\boldsymbol{\psi}}]\mathop{\bigoplus_{{\boldsymbol{n}}=(n_i\geq0)_{i\in I}}}_{\boldsymbol{x}=(x_{ia}\in q^{\mathbb Z})^{i\in I}_{1\leq a\leq n_i}}(\text{Euler characteristic of }N^{\emph{stab}}_{\boldsymbol{x},{\boldsymbol{m}},{\mathbf{l}}})\prod_{i\in I}\prod_{a=1}^{n_i}A_{i,x_{ia}}^{-1}.$$

Here the variety is constructed from the graded quiver with vertex set $I\times\mathbb Z$, arrows $(i,s)\mapsto(j,s-d_{ij})$, and the critical locus of the trace of the potential $W^{\emph{gr}}$, together with the stated generic framed-path vanishing conditions. The conjecture was proved by relating the $q$-characters of $L({\boldsymbol{\psi}})$ to a cluster algebra associated to the quiver $Q^{\emph{gr}}$.