Fock–Goncharov duality conjecture for the $\mathcal{A}_{SL_2,\Sigma_{g,p}}$ cluster variety

Let $\Sigma=\Sigma_{g,p}$, let $Sk^{ta}(\Sigma)$ be the tagged skein algebra, let $U(\Sigma)$ be the upper cluster algebra, and let $\nu:Sk^{ta}(\Sigma)\to U(\Sigma)$ be the natural homomorphism. Let $U(\Sigma)^+$ be the semiring of positive Laurent polynomials and let $\Theta$ denote the proposed canonical theta basis. Fock–Goncharov duality conjecture. (1) The image of tagged bracelets is precisely the set of extremal elements in $U(\Sigma)^+$. In particular, they span the semiring $U(\Sigma)^+$. (2) The homomorphism $\nu:Sk^{ta}(\Sigma)\to U(\Sigma)$ is surjective. These assertions would identify tagged bracelets with the positive extremal basis and establish equality between the tagged skein algebra and the upper cluster algebra; the supplied text does not state a resolution.