Kimura's quantization conjecture for quantum unipotent subgroups

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, let $W$ be its Weyl group, and let $w\in W$. Denote by ${\mathsf{A_q}}[N_{-}(w)]$ the corresponding quantum unipotent subgroup. Quantization conjecture. Up to rescaling by a power of $q$, ${\mathsf{A_q}}[N_{-}(w)]$ is a quantum cluster algebra, and its dual canonical basis contains the quantum cluster monomials. This conjecture formulates the expected relationship between quantum cluster algebras and dual canonical bases. The source states that it is verified in full generality in the paper's abstract, while the supplied candidate has no explicit resolution status; its database status is therefore left open pending verification of the precise result.