Diameter criterion for bounded finite-graph percolation thresholds

Let $G$ be a finite transitive graph, let $|G|$ denote its number of vertices, and let $p^G$ be its percolation threshold. Diameter criterion conjecture. There is a constant $C<1$ such that, whenever

$$\operatorname{diameter}(G)<\frac{|G|}{\log|G|},$$

then $p^G<C$. The paper describes this condition as conjecturally sharp; the general assertion remains open.