The canonical-basis conjecture for quantum cluster monomials

Let $w$ be an element of the Weyl group, let ${\mathcal C}_w$ be the associated category, and let ${\mathcal B}(w)$ be the canonical basis of $\mathcal{A}_{\mathbb{Q}(q)}({\mathcal C}_w)$ obtained by pulling back the relevant subset of the dual canonical basis. A reachable rigid module is a rigid module in ${\mathcal C}_w$ obtainable by the prescribed cluster mutations, and $Y_R$ denotes its associated quantum cluster monomial. Canonical-basis conjecture. Every quantum cluster monomial $Y_R$, where $R$ runs over the set of reachable rigid modules in ${\mathcal C}_w$, belongs to ${\mathcal B}(w)$. Quantum cluster monomials already satisfy one of the characterizing properties of the canonical basis, while the required triangularity condition is not established in the source.