Ferber–McKinley–Samotij's conjecture on nontrivial extremal growth

Let $H$ be a graph that is not a forest. Define its 2-density by

$$m_2(H)=\max\left\{\frac{e(F)-1}{v(F)-2}:F\subseteq H,\ v(F)>2\right\}.$$

Ferber–McKinley–Samotij's conjecture. There exists an $\varepsilon>0$ such that

$$\operatorname{ex}(n,H)\geq \varepsilon n^{2-1/m_2(H)+\varepsilon}.$$

The conjecture expresses the belief that the elementary probabilistic lower bound is not asymptotically tight for graphs containing a cycle. The source presents it as open and uses this type of extremal-growth hypothesis in enumeration results.