  We prove 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 (we call such branchings good pair).
  This is best possible in terms of the arc-connectivity as there are
infinitely many strong digraphs with independence number 2 and arbitrarily high
minimum in-and out-degrees that have good no pair. The result settles a
conjecture by Thomassen for digraphs of independence number 2. We prove that
every digraph on at most 6 vertices and arc-connectivity at least 2 has a good
pair and give an example of a 2-arc-strong digraph $D$ on 10 vertices with
independence number 4 that has no good pair. We also show that there are
infinitely many digraphs with independence number 7 and arc-connectivity 2 that
have no good pair. Finally we pose a number of open problems.
