The second Kahn–Kalai conjecture for graph-containment thresholds

For a graph $H$, let $p_c(H)=p_c(n,H)$ be the unique $p$ such that

$$\mathbb{P}(G_{n,p}\supseteq H)=1/2.$$

Define the expectation threshold by

$$p_{\mathbb E}(H)=p_{\mathbb E}(n,H)=\min\{p:\mathbb{E}_pX_I\geq 1/2\ \forall I\subseteq H\}.$$

The second Kahn–Kalai conjecture. There is a fixed $K$ such that for any graph $H$,

$$p_c(H)<Kp_{\mathbb E}(H)\log v_H.$$

This conjecture asks whether the graph-containment threshold is bounded by a logarithmic factor times the expectation threshold. It was the starting point for the cited work on thresholds, but the general assertion remains open.