Isoperimetric criterion for bounded percolation thresholds

Let $G$ be a finite graph and define

$$\delta(G)=\sup\{\alpha:|\partial S|\geq |S|^\alpha\text{ for all }S\subset G, \ |S|\leq |G|/2\}.$$

Here $\partial S$ is the vertex boundary used in the paper, and $p^G$ is the percolation threshold. Isoperimetric-threshold conjecture. If $\delta(G)>\delta>0$, then there is a function $f$ such that $p^G<f(\delta)<1$. This is proposed as a sharpening of an earlier theorem and is left open.