Minimum Fourier entropy–influence conjecture

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a Boolean function. Its minimum Fourier entropy and total influence are

$$\operatorname{\mathbf{H}}_{\infty}(f)=\min_{S\subset[n]}\log\frac{1}{|\widehat f(S)|^2},\qquad \operatorname{\mathbf{I}}(f)=\sum_{S\subset[n]}|\widehat f(S)|^2|S|.$$

Minimum Fourier entropy–influence conjecture. There exists a constant $c>0$ such that, for every $n$ and every Boolean function $f$,

$$\operatorname{\mathbf{H}}_{\infty}(f)\leq c\operatorname{\mathbf{I}}(f).$$

This is the minimum-entropy analogue of the Fourier entropy–influence conjecture. The supplied excerpt gives no resolution status or additional evidence, so the conjecture is recorded as open.