The worst-case output-size conjecture for the quasi-kernel algorithm

Let $D$ be a sourceless digraph on $n$ vertices, and let Algorithm~ denote the algorithm introduced in the paper, with input consisting of $D$ and a vertex $v$.

Worst-case output-size conjecture. For every $\varepsilon>0$, there exists a sourceless digraph $D$ such that, for every vertex $v$ of $D$, every quasi-kernel returned by the algorithm on inputs $D$ and $v$ has order at least

$$(1-\varepsilon)n.$$

This asks whether the algorithm can have output size arbitrarily close to the whole graph, despite the paper's examples showing only that a small output cannot always be theoretically guaranteed. The supplied text presents this as an open question about worst-case behavior.