Friedgut–Kalai Fourier-Entropy-Influence conjecture for the uniform cube

Let $f:(\{0,1\}^n,\mu_{1/2}^n)\to\{\pm1\}$ be Boolean-valued, with spectral entropy

$${\rm Ent}_{1/2}(f)=\sum_{S\subset[n]}\hat f(S)^2\log\left(\frac{1}{\hat f(S)^2}\right)$$

and total influence ${\rm I}^{(1/2)}[f]=\sum_{i=1}^n{\rm Inf}_i^{(1/2)}[f]$. Friedgut–Kalai's FEI conjecture. There exists a universal constant $C>0$ such that for every $n$ and every Boolean function $f:(\{0,1\}^n,\mu_{1/2}^n)\to\{\pm 1\}$,

$${\rm Ent}_{1/2}(f) \le C\cdot {\rm I}^{(1/2)}[f].$$

This is a longstanding open problem concerning upper bounds for Fourier entropy in terms of influence on the uniform discrete cube.