Leclerc's conjecture on products of dual canonical basis elements

Let $\mathbf{b}_\mu^*$ and $\mathbf{b}_\nu^*$ be two dual canonical basis elements of $\mathcal{A}_q(\operatorname{\mathfrak{n}})$, and suppose that one of them is real. Assume that

$$\mathbf{b}_\mu^*\mathbf{b}_\nu^*\notin q^{\mathbb{Z}}\mathbf{B}^*.$$

Leclerc's conjecture. The product has the form

$$\mathbf{b}_\mu^*\mathbf{b}_\nu^*=q^m\mathbf{b}'^{*}+q^s\mathbf{b}”^*+\sum_{\mathbf{b}^*\neq\mathbf{b}'^*,\mathbf{b}”^*}\gamma_{\mathbf{b}_\mu^*,\mathbf{b}_\nu^*}^{\mathbf{b}^*}(q)\mathbf{b}^*,$$

where $\mathbf{b}'^*\neq\mathbf{b}”^*$, $m<s\in\mathbb{Z}$, and, for every $\mathbf{b}^*\neq\mathbf{b}'^*,\mathbf{b}”^*$,

$$\gamma_{\mathbf{b}_\mu^*,\mathbf{b}_\nu^*}^{\mathbf{b}^*}(q)\in q^{m+1}\mathbb{Z}[q]\cap q^{s-1}\mathbb{Z}[q^{-1}].$$

This conjecture predicts a precise two-term leading structure for products of dual canonical basis elements when the product is not a single basis element up to a power of $q$. The surrounding discussion identifies the classification of real dual canonical bases and equivalent criteria for products lying in $q^{\mathbb{Z}}\mathbf{B}^*$ as difficult open problems.