Embedding a directed graph in its complement

Question

Let $\Gamma$ be a directed graph on $n$ vertices with at most $2n - 4$ edges. Is $\Gamma$ embeddable in its complement?

In other words, does there exist a bijection $\alpha$ of $V(\Gamma)$ such that $(x,y) \in E(\Gamma)$ implies $(\alpha(x), \alpha(y)) \not\in E(\Gamma)$?

This property is known to hold for undirected graphs (on $n$ vertices with at most $n-2$ edges) by Burns and Schuster.

Answer 1

This seems to be a still-open question of Benhocine and Wojda (https://doi.org/10.1002/jgt.3190090305) from 1985 (actually, they conjecture that $\Gamma$ may have up to $2n-3$ edges). The problem was asymptotically resolved by Görlich and Żak (https://doi.org/10.1137/090748056) in 2010, who showed that digraphs with at most $2n-o(n)$ edges are embeddable in their complements.