Multiplicative reachability conjecture for common triangular bases

Let ${{\mathbf L}}$ be a common triangular basis of a quantum cluster algebra. A basis element $b\in{{\mathbf L}}$ is real if $b^{2}\in{{\mathbf L}}$. Multiplicative reachability conjecture. Every real basis element $b\in{{\mathbf L}}$ corresponds to a localized quantum cluster monomial. This generalizes the dual-canonical-basis reachability conjecture and, together with the weaker analog of Leclerc's theorem, would imply the analog of Leclerc's conjecture; it remains open in the paper.