The arc-weighted second neighborhood conjecture

Let $D$ be an arc-weighted digraph without loops or two-cycles. For each vertex $v$, let $\delta_v$ be the second-neighborhood weight minus the first-neighborhood weight, as defined from the arc weights. Arc-weighted second neighborhood conjecture. Every arc-weighted digraph $D$ without loops or two-cycles contains a vertex $v$ such that $\delta_v\geq 0$. The paper proves this conjecture for arc-weighted tournaments, while the general arc-weighted statement remains open.