Approximate Seymour second neighbourhood conjecture

Let $G$ be an oriented graph, and let $N^+_1(G,v)$ and $N^+_2(G,v)$ denote the first and second out-neighbourhoods of $v$.

Approximate Seymour conjecture. For every $\epsilon>0$, every oriented graph $G$ has at least one vertex $v$ satisfying

$$|N^+_2(G,v)| \geq (1-\epsilon)|N^+_1(G,v)|.$$

The paper states this as an equivalent reformulation of Seymour's second neighbourhood conjecture, obtained using lexicographic products; it remains open together with the original conjecture.