Which Boolean functions are most informative?

We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let Xⁿ be i.i.d. Bernoulli(l/2), and let Yⁿ be the result of passing Xⁿ through a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, l}ⁿ → {0,1}, we conjecture that I(b(Xⁿ);Yⁿ) ≤ 1 - H (α). While the conjecture remains open, we provide substantial evidence supporting its validity.