The bipartite extremal-separation conjecture

Let $H$ be a bipartite graph, let $M_H(n)$ denote the maximum running time of the $H$-process, and let $\operatorname{ex}(n,H)$ be the Turán extremal number of $H$. Bipartite extremal-separation conjecture. For every bipartite graph $H$,

$$M_H(n)=o(\operatorname{ex}(n,H)).$$

The conjecture is motivated by the gap between known lower bounds for bipartite infection rules and extremal-number upper bounds. The paper notes that an even stronger polynomial separation might hold, but that stronger claim is not included here.