Thomassen's conjecture on good vertex pairs in arc-strong digraphs

Let $D=(V,A)$ be a digraph. It is $k$-arc-strong if $D\setminus A^{\prime}$ remains strong for every subset $A^{\prime}\subseteq A(D)$ of size at most $k-1$. An out-branching of $D$ is a spanning oriented tree in which every vertex except its root has in-degree one, and an in-branching is a spanning oriented tree in which every vertex except its root has out-degree one. A good $(u,v)$-pair is a pair of arc-disjoint out-branching and in-branching rooted at $u$ and $v$, respectively.

Thomassen's conjecture. There exists an integer $K$ such that every $K$-arc-strong digraph $D=(V,A)$ has a good $(u,v)$-pair for every choice of $u,v\in V$.

This conjecture asks for a uniform arc-connectivity condition guaranteeing compatible branchings with arbitrary prescribed roots. The source gives no resolution, so the conjecture remains open.