The edge-deletion extremal upper-bound conjecture

Let $H$ be an infection rule with at least three edges, let $M_H(n)$ denote the maximum running time of the $H$-process, and let $\operatorname{ex}(n,H-e)$ be the Turán extremal number of $H-e$ for an edge $e\in E(H)$. Edge-deletion extremal conjecture. For every infection rule $H$ with at least three edges,

$$M_H(n)\leq O\left(\min_{e\in E(H)}\operatorname{ex}(n,H-e)\right).$$

This conjecture proposes a general extremal upper bound for maximum running times, beyond the bipartite setting. It remains open, as the paper states the claim as a belief following the discussion of bipartite extremal bounds.