Cox–McGinnis–Marchant conjecture on finiteness of accumulation points

Let ${\mathcal H}$ be a non-trivial hereditary property of graphs, and call $p\in[0,1]$ an accumulation point when the edit-distance function of ${\mathcal H}$ is not locally determined by finitely many CRGs. Cox–McGinnis–Marchant's conjecture. Any non-trivial hereditary property has only finitely many accumulation points. Accumulation points can occur for hereditary properties, including at interior values such as $1/4$, while the supplied source gives no resolution of this general finiteness assertion.