  Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence
number at most 2 and arc-connectivity at least 2 has an out-branching $B^+$ and
an in-branching $B^-$ which are arc-disjoint (such two branchings are called a
{\it good pair}), which settled a conjecture of Thomassen for digraphs of
independence number 2. They also proved that every digraph on at most 6
vertices and arc-connectivity at least 2 has a good pair and gave an example of
a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has
no good pair. They asked for the smallest number $n$ of vertices in a
2-arc-strong digraph which has no good pair. In this paper, we prove that every
digraph on at most 9 vertices and arc-connectivity at least 2 has a good pair,
which solves this problem.
