Li–Médard Lα-norm conjecture for balanced Boolean functions

Show that for every balanced Boolean function f:{0,1}^n→{0,1} and every α∈[1,2], the unsymmetrized Lα norm N_α(f)=∑_{x}(T_p f(x))^α of the noise-stability operator T_p is maximized by a dictatorship function f_0.

Background

Li and Médard proposed a family of conjectures linking noise stability norms to dictatorship optimality, which are known to imply versions of the Courtade–Kumar conjecture. The paper discusses these conjectures and provides structural and local optimality results but not a full resolution.

Establishing the conjecture for 1≤α≤2 in the unsymmetrized form would resolve a strong version of the dictatorship optimality hypothesis.