The "Most informative boolean function" conjecture holds for high noise

We prove the “Most informative boolean function” conjecture of Courtade and Kumar for high noise $ϵ\ge 1\mathrm{/}2-\delta$, for some absolute constant $\delta >0$. Namely, if $X$ is uniformly distributed in $\{0,1{\}}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ϵ$, then, provided $ϵ\ge 1\mathrm{/}2-\delta$, for any boolean function $f$ holds $I(f(X);Y)\le 1-H(ϵ)$. This conjecture was previously known to hold only for balanced functions.