Positivity conjecture for dual canonical basis coefficients and coproducts

Let $\mathbf{C}^\imath$ and $\mathbf{C}$ be the dual canonical bases of $\widetilde{{\mathbf U}}^\imath$ and $\widetilde{{\mathbf U}}$, respectively. For $b\in\mathbf{C}^\imath$, write

$$\imath(b)=\sum_{c\in\mathbf{C}}f_c(v)c,\qquad \widetilde{\Delta}(b)=\sum_{b'\in\mathbf{C}^\imath,c\in\mathbf{C}}g_{b',c}(v)b'\otimes c,$$

and

$$(\imath\otimes1)\circ\widetilde{\Delta}(b)=\sum_{c,c'\in\mathbf{C}}h_{c,c'}(v)c\otimes c'.$$

Positivity conjecture. For every $b\in\mathbf{C}^\imath$, all coefficients $f_c(v)$, $g_{b',c}(v)$ and $h_{c,c'}(v)$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$. In particular, the structure constants of the coproduct for $\widetilde{{\mathbf U}}$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$.

The proposition before this statement proves only membership in $\mathbb{Z}[v^{\frac12},v^{-\frac12}]$. The conjecture strengthens this to positivity, and the source gives no evidence that it has been resolved.