Sullivan's second neighbourhood conjecture

Let $G$ be an oriented graph. For a vertex $v$, let $N^+_2(v)$ be its second out-neighbourhood and let $N^-_1(v)$ be its in-neighbourhood.

Sullivan's conjecture. Every oriented graph contains at least one vertex $v$ such that

$$|N^+_2(v)| \geq |N^-_1(v)|.$$

Sullivan proposed this variation of Seymour's conjecture in a survey on the Caccetta–Häggkvist conjecture. The paper constructs special putative counterexamples and studies highly symmetric cases, but the conjecture remains open.