Finiteness conjecture for accumulation points of edit distance functions

Let $\mathcal{H}$ be a non-trivial hereditary graph property, and call a point an accumulation point if it is a limit point of the edit distance function of $\mathcal{H}$. Finiteness conjecture for accumulation points. Any non-trivial hereditary property has only finitely many accumulation points. The paper presents accumulation points as relatively rare, but leaves this finiteness assertion open.