Gerstenhaber–Giaquinto conjecture on boundary triangular r-matrices

Let $m$ and $n$ be relatively prime positive integers with $m<n$. Let $\mathfrak{p}_{m,n}\subseteq\mathfrak{sl}_n$ be the maximal parabolic subalgebra obtained by deleting the $m$-th simple negative root, and let $r_{m,n}$ denote the corresponding Cremmer–Gervais $r$-matrix. A triangular $r$-matrix has carrier $\mathfrak{p}_{m,n}$ when its carrier Lie algebra is $\mathfrak{p}_{m,n}$. Gerstenhaber–Giaquinto conjecture. The triangular $r$-matrix with carrier $\mathfrak{p}_{m,n}$ is a boundary $r$-matrix and lies in the closure of the $SL_n$-orbit of $r_{m,n}$. This conjecture proposes that the generalized boundary examples associated with relatively prime $m$ and $n$ extend the known generalized Jordanian case and give a boundary analogue of the Belavin–Drinfeld classification.