Rooted non-separating out-branching conjecture

Let $D$ be a digraph on at least $L$ vertices. Write $\lambda(D)$ for its arc-connectivity and $\alpha(D)$ for its independence number, and let $B_s^+$ denote an out-branching rooted at $s$. A non-separating out-branching is an out-branching with the non-separation property studied in the paper. Rooted non-separating out-branching conjecture. There exists an integer $L$ such that every digraph $D$ on at least $L$ vertices with $\lambda(D)\geq 2$ and $\alpha(D)=2$ has a non-separating out-branching $B_s^+$ for every choice of $s\in V$. The source notes that an infinite class of examples has such a branching from every vertex, but does not resolve the conjecture.