Kostochka–Luo–Shan conjecture for quasi-kernels and sinks

Let $D$ be a digraph, let $S(D)$ be the set of sinks of $D$, and let $N^-(S(D))$ be the set of in-neighbors of vertices in $S(D)$. Let $\alpha\geqslant \frac{1}{2}$. A quasi-kernel of $D$ is an independent subset of vertices such that the shortest path from every vertex to it has length at most two.

Kostochka–Luo–Shan conjecture. Every digraph $D$ has a quasi-kernel of size at most

$$\alpha\bigl(|V(D)|+|S(D)|-|N^-(S(D))|\bigr).$$

The case $\alpha=\frac{1}{2}$ was proposed by Kostochka, Luo, and Shan, and the source states that this case is equivalent to the small quasi-kernel conjecture. The displayed formulation is presented as a more general conjecture, whose status is not resolved in the supplied text.