MaxCut for ${\mathrm{M}\mathrm{T}\mathrm{P}}_2$ Covariances

Let $X=(X_1,\dots ,X_n)\in \{0,1{\}}^n$ have a multivariate totally positive (${\mathrm{M}\mathrm{T}\mathrm{P}}_2$) law. We prove that $$\sum _{i<j}\mathbb{E}\left[\mid \mathrm{C}\mathrm{o}\mathrm{v}(X_i,X_j\mid X_{[n]\setminus \{i,j\}})\mid \right]\le n\mathrm{/}2,$$ and more generally a weighted MaxCut inequality for the fully conditioned covariances. As an application, we confirm a conjecture of Allen and O’Donnell on correlation rounding for signed ${\mathrm{M}\mathrm{T}\mathrm{P}}_2$ laws.