Belavin–Drinfeld cohomology conjecture for the Drinfeld–Jimbo r-matrix

Let $\mathfrak{g}$ be a simple Lie algebra, let $G=\operatorname{Ad}(\mathfrak{g})$ be its adjoint group, and let $r_{DJ}$ be the Drinfeld–Jimbo $r$-matrix. For a Belavin–Drinfeld $r$-matrix $r_{BD}$, define $H^{1}_{BD}(G,r_{BD})$ as the set of equivalence classes of Belavin–Drinfeld cocycles, where a cocycle is an element $X\in G(\overline{\mathbb{K}})$ satisfying $X^{-1}\sigma(X)\in C(\overline{\mathbb{K}},r_{BD})$ for every $\sigma\in\operatorname{Gal}(\overline{\mathbb{K}}/\mathbb{K})$, and $C(\overline{\mathbb{K}},r_{BD})$ is the subgroup acting trivially on $r_{BD}$. Belavin–Drinfeld cohomology conjecture. The cohomology $H^{1}_{BD}(G,r_{DJ})$ is trivial if and only if $\mathfrak{g}$ is simply laced, namely of type A, D, or E. This conjectures a precise distinction between the Drinfeld–Jimbo cohomologies of simply laced and non-simply-laced simple Lie algebras; the supplied text gives no resolution status.