Large-order non-separating out-branching conjecture

Let $D$ be a digraph on at least $K$ vertices. Write $\lambda(D)$ for its arc-connectivity and $\alpha(D)$ for its independence number. A non-separating out-branching is an out-branching whose deletion has the non-separation property studied in the paper. Large-order non-separating out-branching conjecture. There exists an integer $K$ such that every digraph $D$ on at least $K$ vertices with $\lambda(D)\geq 2$ and $\alpha(D)=2$ has a non-separating out-branching. The conjecture is motivated by a theorem establishing a bound of $14$ vertices; the source gives no resolution of the proposed existence of some universal $K$.