Half-interval conjecture for accumulation points of edit distance functions

Let $\mathcal{H}$ be a non-trivial hereditary graph property, and let accumulation points mean limit points of its edit distance function. Half-interval conjecture. For any non-trivial hereditary property, either $(0,1/2]$ or $[1/2,1)$ is free of accumulation points. Thus, all accumulation points, if any, must lie in one of the two complementary halves of $(0,1)$. This is posed as a further belief in the paper and remains open.