Monotonicity of arc covariance and correlation in uniformly random DAGs

Let $G=(V,A)$ be a directed acyclic graph, and consider two arcs that are incident on a common node. Let covariance and correlation refer to the corresponding arc-indicator random variables. Monotonicity conjecture. Both covariance and correlation between two arcs incident on a common node are monotonically increasing in modulus. The observed values from complete DAG enumerations support this systematic trend, but the paper gives no formal proof.