Multiplicative reachability conjecture for dual canonical bases

Let $w$ be a Weyl group element, let ${\mathsf{A_q}}[N_{-}(w)]$ be the corresponding quantum unipotent subgroup, and let $\mathsf{B}^{{\mathsf{up}}}(w)$ be its dual canonical basis. A basis element $b\in\mathsf{B}^{{\mathsf{up}}}(w)$ is real if $b^{2}\in q^{\mathbb{Z}}\mathsf{B}^{{\mathsf{up}}}(w)$. Multiplicative reachability conjecture. Every real element $b\in\mathsf{B}^{{\mathsf{up}}}(w)$ corresponds, after rescaling, to a quantum cluster monomial. This conjecture would extend the known weaker result from quantum cluster monomials to all real dual canonical basis elements and would imply Leclerc's conjecture for the quantum unipotent subgroup; it remains open in the paper.