Cox–McGinnis–Marchant conjecture on accumulation-point-free halves

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. Either $(0,1/2]$ or $[1/2,1)$ is free of accumulation points for ${\mathcal H}$. Known examples show that interior accumulation points can occur, whereas the supplied source does not resolve whether one of these two half-intervals must always be free of accumulation points.