Matsuda's weakly closed graph characterization of F-purity

Let $G$ be a simple graph on $n$ vertices, let $\Bbbk$ be a field, and set $R=\Bbbk[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$. For $\{i,j\}\in E(G)$, write $f_{i,j}=x_{i}y_{j}-x_{j}y_{i}$, define the binomial edge ideal

$$J_G=(\{f_{i,j}\mid \{i,j\}\in E(G)\})\subseteq R,$$

and set $R_G=R/J_G$. A graph is weakly closed as in Matsuda's definition. Assume that the characteristic of $\Bbbk$ is two. Matsuda's conjecture. $G$ is weakly closed if and only if $R_G$ is F-pure. Matsuda proved the forward implication, while the converse was conjectured and is proved in the source paper; thus the conjecture is solved.