Jayanthan–A R conjecture on linear type binomial edge ideals

Let $G$ be a graph, let $S=\mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_n]$, and let

$$J_G=(x_i y_j-x_j y_i\mid i<j,\ \{i,j\}\in E(G))\subset S$$

be its binomial edge ideal. An ideal $I\subset A$ is of linear type when its symmetric algebra is naturally isomorphic to its Rees algebra, $\operatorname{Sym}(I)\cong\mathcal{R}(I)$. Jayanthan–A R conjecture. If $G$ is a tree or a unicyclic graph, then $J_G$ is of linear type. This conjecture proposes a combinatorial characterization of a class of binomial edge ideals analogous to the known characterization for ordinary edge ideals; its resolution status is not specified in the source.