Friedgut–Kalai entropy/influence conjecture

Let $f:0,1n_{1/2}\to\{-1,1\}$ be a Boolean function on the discrete cube with uniform measure $\mu_{1/2}$. Write $\operatorname{Ent}_{1/2}(f)$ for the entropy of its Fourier-weight distribution and $\operatorname{I}_{1/2}(f)$ for its total influence. Friedgut–Kalai's conjecture. There exists a universal constant $c$ such that, for every $n$ and every such $f$,

$$\operatorname{Ent}_{1/2}(f)\leq c\,\operatorname{I}_{1/2}(f).$$

If true, this would imply strong lower bounds on total influence for graph properties and consequences for Fourier concentration of polynomial-size DNF formulas. The paper states that the conjecture was still wide open, with the best cited graph-property lower bound being $\Omega((\log n)^{2-\epsilon})$ for every $\epsilon>0$.