The Hellinger conjecture for Boolean functions

Let $\mathbf{X}\sim\operatorname{Unif}(\mathbb{H}^n)$ and let $\mathbf{Y}$ be obtained by passing each bit of $\mathbf{X}$ through a binary symmetric channel with correlation parameter $\rho$. Let $f:\mathbb{H}^n\mapsto\mathbb{H}$ be Boolean, and define the noise operator by

$$(T_\rho f)(\mathbf{y})=\mathbb{E}[f(\mathbf{X})\mid\mathbf{Y}=\mathbf{y}].$$

The Hellinger conjecture. Under these assumptions,

$$\sqrt{1-\mathbb{E}[f(\mathbf{X})]^2}-\mathbb{E}\left[\sqrt{1-(T_\rho f)(\mathbf{Y})^2}\right]\leq 1-\sqrt{1-\rho^2}.$$

The conjecture is stronger than the most-informative Boolean function conjecture and would imply it. It is presented as open in the paper.