The most-informative Boolean function conjecture

Let $\mathbf{X}\sim\operatorname{Unif}(\mathbb{H}^n)$ be uniformly distributed on the Boolean hypercube, and let $\mathbf{Y}$ be obtained by passing each bit of $\mathbf{X}$ through a binary symmetric channel with crossover probability $p=(1-\rho)/2\in[0,1]$. Let $f:\mathbb{H}^n\mapsto\mathbb{H}$ be a Boolean function, and let $H_2(p)=-p\log_2(p)-(1-p)\log_2(1-p)$ be binary entropy. The most-informative Boolean function conjecture. One has

$$I(f(\mathbf{X});\mathbf{Y})\leq 1-H_2(p).$$

This conjecture asks which Boolean functions maximize the mutual information through a binary symmetric channel. It is known for balanced Boolean functions in a neighborhood of zero correlation, including $|\rho|\leq 0.44$, but remains open in general.