Gekhtman–Shapiro–Vainshtein correspondence conjecture for simple Lie groups

A simple complex Lie group is a group $\mathcal{G}$ whose Lie algebra is simple. A Belavin–Drinfeld triple is a triple $T=(\Gamma_{1},\Gamma_{2},\gamma)$ consisting of two subsets of the simple roots and a nilpotent isometry between them. A cluster structure $\mathcal{C}_{T}$ is the cluster structure associated with $T$.

Gekhtman–Shapiro–Vainshtein correspondence conjecture. Let $\mathcal{G}$ be a simple complex Lie group. For any Belavin–Drinfeld triple $T=(\Gamma_{1},\Gamma_{2},\gamma)$ there exists a cluster structure $\mathcal{C}_{T}$ on $\mathcal{G}$, and these cluster structures correspond to the Belavin–Drinfeld classification of solutions of the classical Yang–Baxter equation.

The conjecture proposes that the Belavin–Drinfeld classification of classical Yang–Baxter solutions parametrizes cluster structures on simple Lie groups. The paper's abstract states that the conjecture is confirmed for $SL_{5}$, while the supplied span gives only the existence assertion and does not include the remaining conditions or a general resolution.