Hernandez–Leclerc adaptable-word monoidal categorification conjecture

Let $\mathbf i$ be an adaptable word, let $\mathcal A^\dagger(\mathbf i)$ be the associated quantum cluster algebra, and let $\kappa$ be the map to the relevant Grothendieck ring with simple-module basis $\{S(w)\}_{w\in\mathbb N^l}$. Monoidal categorification conjecture. The quantum cluster monomials of $\mathcal A^\dagger(\mathbf i)$ are contained in $\kappa^{-1}(\{S(w)\}_{w\in\mathbb N^l})$. This is the adaptable-word form of monoidal categorification, including the level-$N$ cases mentioned in the source; no resolution evidence is supplied for the stated generality.