The threshold conjecture for biased Maker–Breaker $H$-factor games

Let $b_\Delta$ denote the threshold bias for the biased Maker–Breaker $H$-factor game, where $H$ has maximum degree $\Delta$. Threshold conjecture. For all $\Delta \ge 3$,

$$b_\Delta = \Theta(n^{2/(\Delta + 3)}).$$

This proposes that, up to constant factors, constructing an $H$-factor is no harder for Maker than constructing a $K_{\Delta+1}$-factor; the source provides the $K_r$-factor threshold for $r\ge 4$ as motivation, but does not establish the conjecture for general bounded-degree graphs.