Matsuda's eventual F-purity conjecture for binomial edge ideals

Let $G$ be a graph, let $\Bbbk$ be a field, and let $R_G$ denote the quotient by the binomial edge ideal associated to $G$. Write $p$ for the characteristic of the base field. Matsuda's conjecture. $R_G$ is F-pure for all sufficiently large primes $p>0$. The conjecture is false, as stated explicitly in the source.