Kahn–Kalai optimal-scale conjecture

Let $X$ be a finite set of size $n$, let ${\cal F}\subseteq2^X$ be monotone, and let $p_c({\cal F})$ satisfy $\mu_{p_c({\cal F})}({\cal F})=1/2$. Call ${\cal F}$ $(C,p)$-optimal when

$$p\,m'(p)\leq C m(p)\log_p m(p),\qquad m(p)=\mu_p({\cal F}).$$

Optimal-scale conjecture. For each $C>0$ there is an $\varepsilon>0$ such that, for every monotone ${\cal F}$, there exists

$$p\in\left[\frac{\varepsilon p_c({\cal F})}{\log n},p_c({\cal F})\right]$$

for which ${\cal F}$ is $(C\log(1/p),p)$-optimal. The paper says this conjecture, together with part (c) of its isoperimetric-structure conjecture, would imply the main expectation-threshold conjecture; it was not known whether the asserted scale exists.