Amicability conjecture for vertex pairs in networks

Let $G$ be a network, and let $v\neq w$ be vertices of $G$. The paper defines the property that the pair $\{v,w\}$ is amicable in $G$ through its associated polynomial interlacing conditions.

Amicability conjecture. For every network $G$ and every pair of distinct vertices $v$ and $w$, the pair $\{v,w\}$ is amicable in $G$.

This conjecture is stated as a strengthening that would imply the paper’s earlier conjecture. The source proves the property for series-parallel networks and notes related results for cactus networks, but does not resolve it for arbitrary networks.