The edge-deletion conjecture for upper thresholds

Let $H$ be a graph with at least three edges. For an edge $e$ of $H$, let $H-e$ be the graph obtained by deleting $e$, together with a vertex of degree one if that vertex is an endpoint of $e$. Let $f^+_H(n)$ and $f^-_{H-e}(n)$ denote the upper and lower thresholds of the strict $(1\colon b)$ Avoider–Enforcer games. Edge-deletion conjecture. For every graph $H$ with at least three edges, one has

$$f^+_H(n)=\Theta\big(\max\limits_{e\in H} f^-_{H-e}(n)\big).$$

The conjecture proposes that the upper threshold is governed by the largest lower threshold among the edge-deleted graphs; the source presents it as an open problem for general graphs.