Weyl-invariant square-root cluster functions conjecture

Let $\mathcal X_{|\mathcal A_n|}$ be the cluster variety associated with the $\mathcal A_n$-quiver, let $O(\sqrt{\mathcal X_{|\mathcal A_n|}})$ be the ring of universal Laurent polynomials in square roots of cluster variables, let $W$ be the birational Weyl group acting on this ring, and let $\mathbf G_n$ be the ring generated by formal geodesic functions. Weyl-invariant square-root functions conjecture.

$$O(\sqrt{\mathcal X_{|\mathcal A_n|}})^W=\mathbf G_n.$$

The preceding theorem proves the inclusion $O(\mathcal X_{|\mathcal A_n|})^W\subset\mathbf G_n$; the conjecture asks whether adjoining square roots produces no additional Weyl invariants beyond the formal geodesic functions. The source does not state a resolution.