Laurent positivity conjecture for cluster ensemble invariant bases

Let $Q$ be the quiver defining the cluster ensemble with $\mathcal A$-space and $\mathcal X$-space, and let $\Gamma$ act on these spaces with subgroup $\Gamma_\circ$. A correspondence basis is a pair of bases for the invariant fields whose elements correspond via denominator vectors. Laurent positivity conjecture. There is a basis of $\mathbb{R}(\mathcal{X}_Q)^{\Gamma_\circ}$ with a corresponding basis of $\mathbb{R}(\mathcal{A}_Q)^{\Gamma_\circ}$ such that $\Gamma$ acts on the basis of $\mathbb{R}(\mathcal{A}_Q)^{\Gamma_\circ}$ by positive Laurent polynomials. This conjecture concerns the expected Laurent structure of invariant functions and is presented as evidence for a correspondence between the $\mathcal A$- and $\mathcal X$-invariants; the source does not establish it in general.