Dense fixed-subgraph conjecture for the final H-free process

Let $H$ be a strictly $2$-balanced graph, and let $d_2(H)$ denote its $2$-density. Let $F$ be a fixed non-empty graph satisfying

$$m(F)>d_2(H).$$

Dense fixed-subgraph conjecture. With high probability, the final graph of the $H$-free process contains no copy of $F$. This strengthens the proved result for graphs whose number of vertices may grow moderately with $n$, and is motivated by the fact that the known bound for the maximum density of fixed graphs appearing in the process is best possible up to constants. The statement is presented as a belief in the source, and no resolution is supplied here.