The large-quasikernel structural conjecture

Let $D$ be a finite digraph. A quasikernel is an independent set from which every vertex is reachable by a directed path of length at most $2$. For a vertex $y$, let $\deg^+(y)$ denote its out-degree.

Large-quasikernel structural conjecture. Every digraph $D$ either contains a subset $A\subseteq V(D)$ such that $D[A]$ is acyclic and

$$|A|\ge |V(D)|^{1/2},$$

or contains a vertex $y\in V(D)$ that belongs to some quasikernel and satisfies

$$\deg^+(y)\ge |V(D)|^{1/2}-1.$$

If true, this structural dichotomy would improve the paper's lower bounds on the size of the closed out-neighborhood of a quasikernel. The source presents it as a conjectural route toward the Large Quasikernel Conjecture.