Maximal asymmetric alpha-stability conjecture

Let $\rho$ satisfy $0\leq\rho\leq1$, let $b1\in[1,9]$, and let $\mathbf{MaxStab}_{\alpha}(1/2)$ denote the maximal asymmetric $b1$-stability at mean $1/2$. A dictator function is a Boolean function depending on one coordinate. Maximal asymmetric alpha-stability conjecture. For $0\leq\rho\leq1$ and $b1\in[1,9]$, $\mathbf{MaxStab}_{\alpha}(1/2)$ is attained by dictator functions. The conjecture unifies and slightly generalizes the Mossel–O'Donnell, Courtade–Kumar, and Li–Médard conjectures; according to the supplied status evidence, it remains open for $b1\in[1,2)\cup(2,9]$.