The out-degree-three expansion conjecture for quasi-kernels

Let $D$ be a sourceless oriented graph, and write $N^+(v)$ for the out-neighborhood of a vertex $v$ and $S(v)$ for the set used in the paper's preceding theorem. Assume that $D$ has maximum out-degree at most $3$.

Out-degree-three expansion conjecture. There exists a vertex $v\in V(D)$ such that

$$|N^+(v)\cup N^+(S(v))|\geq |S(v)|+1.$$

The conjecture is proposed as a strengthening of the proof of the paper's out-degree-three theorem and would provide a local structural condition useful for constructing small quasi-kernels. The supplied text does not establish it or give evidence of resolution.