The upper-threshold conjecture for unicyclic graphs

Let $H$ be a unicyclic graph, meaning a connected graph containing exactly one cycle, and let $v(H)$ denote its number of vertices. Let $f^+_H(n)$ be the upper threshold of the strict $(1\colon b)$ Avoider–Enforcer $H$-game on $K_n$. Upper-threshold conjecture for unicyclic graphs. For every unicyclic graph $H$, one has

$$f^+_{H}(n)=\Theta\big(n^{\frac{v(H)}{v(H)-1}}\big).$$

The stated order is known for infinitely many values of $n$, while the conjecture asserts it for all $n$; the problem remains open even for a triangle.