Axiom for the small quasi-kernel conjecture

Let $D$ be a non-empty digraph. For a vertex $u\in V(D)$, let $d^+(u)$ denote its out-degree, let $N^+[u]$ denote its closed out-neighborhood, and call a vertex a source if it has in-degree zero. Axiom for the small quasi-kernel conjecture. There exists a vertex $u\in V(D)$ such that $d^+(u)>0$ and $D-N^+[u]$ has at most $d^+(u)$ sources not present in $D$. If this axiom holds for every directed graph, the paper's preceding argument proves the small quasi-kernel conjecture, namely that every source-free directed graph has a quasi-kernel of size at most half its vertices.