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.
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