Exotic cluster structures on $SL_n$ with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin--Drinfeld classification of solutions to the classical Yang Baxter equation. For any non trivial Belavin--Drinfeld data of minimal size for $SL_{n}$, the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin--Drinfeld data. This paper proves the rest of the conjecture: the corresponding upper cluster algebra $\overline{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}\left(SL_{n}\right)$, the torus determined by the BD triple generates theaction of $(\mathbb{C}^{*})^{2k_{T}}$ on $\mathbb{C}\left(SL_{n}\right)$, and the correspondence between Belavin--Drinfeld classes and cluster structures is one to one.