The weak-closedness characterization of F-purity for binomial edge ideals in characteristic 2

Let $G$ be a graph, let $S$ be the polynomial ring associated with $G$, and let $J_G$ be its binomial edge ideal. Assume that $p=2$.

Weak-closedness characterization. The quotient $S/J_G$ is $F$-pure if and only if $G$ is weakly closed.

The preceding theorem proves the implication from weak closedness to $F$-purity for arbitrary $p$. The converse, and hence the equivalence in characteristic $2$, was verified in the source for graphs with at most six vertices using Macaulay2, but remains conjectural in general.