Courtade–Kumar conjecture on information-maximizing Boolean functions

Let $\mathcal{C}_n=\{-1,1\}^n$ be the Boolean hypercube, let $X$ be uniformly distributed on $\mathcal{C}_n$, and let $\mathcal{N}_{X,\rho}$ denote the corresponding noisy version of $X$. For a Boolean function $f:\mathcal{C}_n\to\{0,1\}$, write $\mathrm{I}(X;f(\mathcal{N}_{X,\rho}))$ for the mutual information between $X$ and $f(\mathcal{N}_{X,\rho})$. Courtade–Kumar conjecture. Among all Boolean functions $f:\mathcal{C}_n\to\{0,1\}$, the function maximizing

$$\mathrm{I}(X;f(\mathcal{N}_{X,\rho}))$$

is the dictator function. This is an entropic analogue of Boolean noise-stability extremal results; resolving it would identify dictators as the most informative Boolean observations of a noisy hypercube point. The conjecture is attributed to Courtade and Kumar, and no resolution is indicated in the supplied text.