Leclerc's conjecture on products of real dual canonical basis elements

Let ${\mathfrak{g}}$ be a Kac--Moody algebra with symmetrizable Cartan datum, let $\mathsf{U_q}^{-}$ be the negative part of its quantized enveloping algebra, and let $\mathsf{B}^{{\mathsf{up}}}$ be its dual canonical basis. A basis element $b\in\mathsf{B}^{{\mathsf{up}}}$ is real if $b^{2}\in q^{\mathbb{Z}}\mathsf{B}^{{\mathsf{up}}}$. Leclerc's conjecture. Assume that $b_{1}$ is real. For any $b_{2}\in\mathsf{B}^{{\mathsf{up}}}$ such that $b_{1}b_{2}\notin q^{\mathbb{Z}}\mathsf{B}^{{\mathsf{up}}}$, the expansion of their product on $\mathsf{B}^{{\mathsf{up}}}$ has the form

$$b_{1}b_{2}=q^{h}b'+q^{s}b”+\sum_{c\neq b',b”}\gamma_{b_{1},b_{2}}^{c}c,$$

where $b'\neq b”$, $h<s\in\mathbb{Z}$, and $\gamma_{b_{1},b_{2}}^{c}\in q^{h+1}\mathbb{Z}[q]\cap q^{s-1}\mathbb{Z}[q^{-1}]$. This conjecture describes a crystal-operator-like multiplication rule for dual canonical bases; its status in the paper is not resolved.