The lower-threshold conjecture for Avoider–Enforcer games

For a non-empty graph $H$, define

$$m(H)=\max\limits_{F\subseteq H:\,v(F)\ge 1}\frac{e(F)}{v(F)},$$

where $v(F)$ and $e(F)$ are the numbers of vertices and edges of $F$, respectively. Let $f^-_H(n)$ be the lower threshold of the strict $(1\colon b)$ Avoider–Enforcer $H$-game on $K_n$. Lower-threshold conjecture. For every graph $H$ with at least two edges, one has

$$f^-_H(n)=\Theta\big(n^{\frac{1}{m(H)}}\big).$$

The paper proves this order when $m(H)\leq 1$ and conjectures it for all graphs with at least two edges; the missing part is a suitable winning strategy for Enforcer.