Symmetrized Li–Médard conjecture

Let $X$ be uniformly distributed on $\{0,1\}^n$, let $Y$ be obtained from $X$ through a binary symmetric channel with crossover probability $0<p<\frac{1}{2}$, and let $f:\{0,1\}^n\to\{0,1\}$ be balanced, meaning $\mathbb{E}[f(Y)]=\frac{1}{2}$. Define $T_pf(x)=\mathbb{P}(f(Y)=1\mid X=x)$, let $f_0$ be a dictatorship function, and set

$$N^\mathsf{sym}_\alpha(f)=\sum_{x\in\{0,1\}^n}\left((T_pf(x))^\alpha+(1-T_pf(x))^\alpha\right).$$

A Li–Médard conjecture. For any balanced $f$,

$$N^\mathsf{sym}_\alpha(f)\leq N^\mathsf{sym}_\alpha(f_0)$$

for $1\leq\alpha\leq2$. This symmetrized conjecture is the formulation linked in the paper to the mutual-information version of the Courtade–Kumar conjecture.