The Large Quasikernel Conjecture

Let $D$ be a finite digraph. A quasikernel is an independent set $Q\subseteq V(D)$ such that every vertex $x\in V(D)$ satisfies $\operatorname{dist}(Q,x)\le 2$. For $S\subseteq V(D)$, its closed out-neighborhood is

$$N^+[S]=S\cup\{v:\exists u\in S,\ uv\in E(D)\}.$$

Large Quasikernel Conjecture. Every digraph $D$ contains a quasikernel $Q$ with

$$|N^+[Q]|\ge\frac{1}{2}|V(D)|.$$

This is a weak form of the Small Quasikernel Conjecture in terms of the portion of the digraph covered directly by a quasikernel. The source reports partial progress but no resolution.