Berenstein–Greenstein positivity conjecture for coproduct structure constants

Let $\widetilde{{\mathbf U}}$ be the quantum group with double (equivalently, dual) canonical basis, and let its coproduct be the corresponding bialgebra coproduct. Berenstein–Greenstein's coproduct positivity conjecture. The structure constants of the double (dual) canonical basis with respect to the coproduct also lie in

$$\mathbb{N}[v^{\frac{1}{2}},v^{-\frac{1}{2}}].$$

This would extend the known positivity of coproduct structure constants for Lusztig's canonical basis of ${\mathbf U}^+$ to the entire quantum group. The paper proposes this statement after identifying the double canonical basis with the dual canonical basis; its resolution is not given in the supplied text.