Friedgut–Kalai sharp-threshold conjecture for monotone graph properties

Let $P$ be a monotone property of graphs on $n$ vertices, let $u_p^n$ denote the product measure on graphs in which each edge is present with probability $p$, and let $u_q^n$ be defined similarly. For $epsilon>0$, assume

$$q=p+c\frac{\log(1/2\epsilon)}{\log^2 n}.$$

Friedgut–Kalai conjecture. There is an absolute constant $c$ such that, if $u_p^n(P)>\epsilon$, then

$$\nu_q^n(P)>1-\epsilon.$$

This is a proposed improvement of the Friedgut–Kalai sharp-threshold bound, replacing the denominator $\log n$ by $\log^2 n$ for arbitrary monotone graph properties. The supplied source does not state whether the question has been resolved.