The minimum in-degree conjecture for small quasikernels

Let $D$ be a finite digraph. A quasikernel is an independent set $Q\subseteq V(D)$ such that every vertex is reachable from $Q$ by a directed path of length at most $2$. The minimum in-degree of $D$ is denoted by $\delta^-(D)$.

Minimum in-degree quasikernel conjecture. There exists an integer $\delta\ge 2$ such that every digraph $D$ with $\delta^-(D)\ge\delta$ contains a quasikernel $Q$ with

$$|Q|\le\frac{1}{3}|V(D)|.$$

This conjecture proposes that sufficiently large minimum in-degree forces a quasikernel substantially smaller than the general one-half bound. The source states that the behavior for general minimum in-degree remains unclear.