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We prove that every digraph of independence number at most 2 andarc-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 areinfinitely many strong digraphs with independence number 2 and arbitrarily highminimum in-and out-degrees that have good no pair. The result settles aconjecture by Thomassen for digraphs of independence number 2. We prove thatevery digraph on at most 6 vertices and arc-connectivity at least 2 has a goodpair and give an example of a 2-arc-strong digraph $D$ on 10 vertices withindependence number 4 that has no good pair. We also show that there areinfinitely many digraphs with independence number 7 and arc-connectivity 2 thathave no good pair. Finally we pose a number of open problems.

Arc-disjoint in- and out-branchings in digraphs of independence number at most 2