Bang-Jensen and Wang's conjecture on branchings in semicomplete split digraphs

Let $D$ be a semicomplete split digraph, meaning a split digraph whose relevant semicomplete structure has no pair of non-adjacent vertices. It is $2$-arc-strong if deleting any one arc leaves a strong digraph. A good $(u,v)$-pair is a pair of arc-disjoint out-branching and in-branching rooted at $u$ and $v$, respectively.

Bang-Jensen and Wang's conjecture. Every $2$-arc-strong semicomplete split digraph $D$ contains a good $(u,v)$-pair for every choice of vertices $u,v$ in $D$.

This conjecture narrows the general branching problem to semicomplete split digraphs. The source describes related counterexamples for general $2$-arc-strong split digraphs but gives no resolution of this semicomplete case.