Two-term expansion conjecture for products involving a real dual canonical basis vector

Let $b_1,b_2\in{\bf B}^*$, where $b_1$ is real, meaning $b_1^2\in q^{\mathbb Z}{\bf B}^*$. Suppose that $b_1b_2\notin q^{\mathbb Z}{\bf B}^*$. Two-term expansion conjecture. The expansion of $b_1b_2$ in ${\bf B}^*$ has the form

$$b_1b_2=q^m b'+q^s b”+\sum_{c\ne b',b”}\gamma_{b_1b_2}^c(q)c,$$

where $b'\ne b”$, $m,s\in\mathbb Z$, $m<s$, and $\gamma_{b_1b_2}^c(q)\in\mathbb Z[q,q^{-1}]$; moreover, whenever $\gamma_{b_1b_2}^c(q)\ne0$,

$$\gamma_{b_1b_2}^c(q)\in q^{m+1}\mathbb Z[q]\cap q^{s-1}\mathbb Z[q^{-1}].$$

The conjecture is motivated by the behavior of products with a real basis vector and is checked in the paper when $b_1$ is a Chevalley generator, but no resolution is supplied for the general case.