Positivity conjecture for generalized quantum cluster algebras

Let $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ be a generalized quantum cluster algebra with initial extended cluster $\{X_1,\ldots,X_m\}$. A Laurent polynomial in this cluster has the form $f(X_1^{\pm1},\ldots,X_m^{\pm1})$, and $\mathbb{N}[q^{\pm\frac{1}{2}},h_{i,r}(q^{\frac{1}{2}})]$ denotes the coefficient semiring generated by the indicated parameters. Positivity conjecture. Any cluster variable $X$ of $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ can be expressed as a Laurent polynomial $f(X_1^{\pm1},\ldots,X_m^{\pm1})$ in the initial extended cluster $\{X_1^{\pm1},\ldots,X_m^{\pm1}\}$ with coefficients belonging to $\mathbb{N}[q^{\pm\frac{1}{2}},h_{i,r}(q^{\frac{1}{2}})]$. The explicit type $G_2$ expansions preceding the conjecture exhibit this positivity in examples; the conjecture asserts it for every cluster variable of every generalized quantum cluster algebra, while the general claim is not resolved in the supplied source.