The analog of Leclerc's conjecture for common triangular bases

Let ${{\mathbf L}}$ be a common triangular basis of a quantum cluster algebra. A basis element $R\in{{\mathbf L}}$ is real if $R^{2}\in{{\mathbf L}}$. The analog of Leclerc's conjecture. For any real basis element $R\in{{\mathbf L}}$ and any $V\in{{\mathbf L}}$, either

$$R*V\in v^{\mathbb{Z}}{{\mathbf L}},$$

or

$$R*V=v^{s}S+\sum_{j}b_{j}L^{(j)}+v^{h}H,$$

where $s>h\in\mathbb{Z}$, $b_{j}\in v^{h+1}\mathbb{Z}[v]\cap v^{s-1}\mathbb{Z}[v^{-1}]$, and $S$, $L^{(j)}$, and $H$ are finitely many distinct elements of ${{\mathbf L}}$. This is proposed as an analog of Leclerc's multiplication conjecture after replacing the dual canonical basis by the common triangular basis; a weaker form is proved for localized quantum cluster monomials, but the full assertion remains open.