  Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of
$D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus
K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved
that every digraph has a quasi-kernel. P. L. Erd\H{o}s and L. A. Sz\'ekely in
1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$
has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed
for narrow classes of digraphs, such as semicomplete multipartite,
quasi-transitive, or locally demicomplete digraphs. In this note, we state a
similar conjecture for all digraphs, show that the two conjectures are
equivalent, and prove that both conjectures hold for a class of digraphs
containing all orientations of 4-colorable graphs (in particular, of all planar
graphs).
