Expectation of Gaussian r.v. conditioned on positive r.v.s with positive covariances is positive

Question

Suppose that $(X_1,\dotsc,X_K)^T \sim \mathcal{N}(0, \Sigma)$, with $\mathrm{cov}(X_i, X_j) > 0$ for all $i,j$. Prove that $$ \mathbb{E}[X_K 1\{ X_1> 0, \dotsc, X_{K-1}> 0 \} ] > 0$$ where $1\{ A \}$ is the indicator function of $A$.

Answer 1

This follows immediately from the more general result in

Pitt, Loren D. "Positively correlated normal variables are associated." The Annals of Probability (1982): 496-499.