Berenstein–Zelevinsky's positivity conjecture for quantum cluster algebras

Let $\mathcal A_v$ be a quantum cluster algebra and let $t$ be a seed with quantum cluster variables $X(t)$. For ${\bf a}\in\mathbb Z^m$, let $X(t)^{{\bf a}}$ denote the associated quantum torus monomial.

Berenstein–Zelevinsky's positivity conjecture. For any quantum cluster variable $X$ of $\mathcal A_v$,

$$X\in \mathbb N[v^{\pm1}]\langle X(t)^{{\bf a}}\mid {\bf a}\in \mathbb Z^m\rangle.$$

This is the positivity form of the quantum Laurent phenomenon: the Laurent expansion in the variables of any seed has coefficients in $\mathbb N[v^{\pm1}]$. The source attributes the conjecture to Berenstein and Zelevinsky; the supplied text does not state a resolution for the conjecture in this general formulation.