Isomorphism conjecture for twisted Lie bialgebra structures corresponding to Belavin–Drinfeld matrices

Let $r_{BD}$ be an $r$-matrix from the list in Lemma, and let $\mathfrak{g}$ be the corresponding Lie algebra. Twisted Lie bialgebra structures on $\mathfrak{g}$ corresponding to $r_{BD}$ are the objects under consideration. Isomorphism conjecture. If $r_{BD}$ is an $r$-matrix from the list of Lemma, then all twisted Lie bialgebra structures on $\mathfrak{g}$ corresponding to $r_{BD}$ are isomorphic. This conjecture is motivated by the comparison between the orthogonal and special linear cases, where non-conjugate twisted structures can nevertheless be isomorphic. The supplied text gives no resolution, so the conjecture remains open.