Quantitative forcing conjecture for bipartite graphs

Let $H$ be a fixed bipartite graph of girth $g$, and let $p$ be a fixed constant with $0<p<1$. For each $\epsilon>0$, there is a parameter $\delta=\Omega(\epsilon^g)$ such that any graph $G$ of density $p$ satisfying $\mathcal{P}_{H,p}(\delta)$ also satisfies $\mathcal{P}_{2,p}^*(\epsilon)$. Quantitative forcing conjecture. The conclusion holds with a forcing parameter of order at least $\epsilon^g$. This is stated as a stronger quantitative version of the forcing conjecture; the source gives no resolution, so the proposed dependence remains open.