Bang-Jensen and Gutin's algorithmic conjecture for good pairs in quasi-transitive digraphs

Let $D=(V,A)$ be a quasi-transitive digraph, meaning that whenever $uv$ and $vw$ are arcs, there is an arc between $u$ and $w$. For vertices $u,v\in V$, a good $(u,v)$-pair is a pair of arc-disjoint branchings consisting of an out-branching rooted at $u$ and an in-branching rooted at $v$.

Bang-Jensen and Gutin's conjecture. There exists a polynomial algorithm for deciding, given a quasi-transitive digraph $D=(V,A)$ and two vertices $u,v$ of $V$, whether $D$ has a good $(u,v)$-pair.

The conjecture asks for an efficient recognition algorithm extending the known result for good $(u,u)$-pairs in quasi-transitive digraphs. Its status is not resolved in the supplied source.