Full Fock–Goncharov conjecture for a cluster variety

Let $V$ be a cluster variety, let $V^{\vee}(\mathbb{Z}^t)$ be the tropical integral points on its dual cluster variety, let $\Theta$ be the proposed theta-function index set, and let $\nu'$ be the associated homomorphism. Write $\mathrm{mid}(V)$, $\mathrm{can}(V)$, and $\mathrm{up}(V)$ for the middle, canonical, and upper cluster algebras, respectively. Full Fock–Goncharov conjecture. (1) The homomorphism $\nu'$ is injective. (2) The set $\Theta$ exhausts the tropical points $V^{\vee}(\mathbb{Z}^t)$, namely $\mathrm{mid}(V)=\mathrm{can}(V)$. (3) The homomorphism $\nu'$ is surjective, equivalently $\nu'(\mathrm{can}(V))=\mathrm{up}(V)$. Together these assertions identify the theta basis with the canonical basis and the upper cluster algebra; the supplied text does not state a resolution.