Talagrand's fractional expectation-threshold conjecture

Let $X$ be a finite set and let ${\mathcal F}\subseteq 2^X$ be increasing. Let $p_c({\mathcal F})$ be the threshold defined by $\mu_p({\mathcal F})=1/2$. Define the fractional expectation-threshold $q_f({\mathcal F})$ as the largest $p$ for which there is a function $g:2^X\to\mathbb R^+$ satisfying

$$\sum_{S\subseteq T}g(S)\geq1\quad\text{for every }T\in{\mathcal F},\qquad \sum_{S\subseteq X}g(S)p^{|S|}\leq\frac12.$$

Talagrand's fractional expectation-threshold conjecture. There is a universal constant $K$ such that

$$p_c({\mathcal F})\leq Kq_f({\mathcal F})\log|X|.$$

This is a fractional relaxation of the Kahn–Kalai expectation-threshold conjecture. The source presents it as a conjecture of Talagrand and notes that a stronger theorem replacing $|X|$ by a smaller parameter is proved in the paper.