Arc-disjoint out-branching and in-branching in highly arc-connected digraphs

Determine whether there exists a natural number K such that every K-arc-strong digraph contains an out-branching and an in-branching that are arc-disjoint.

Background

The paper defines a digraph as K-arc-strong if it remains strongly connected after deleting any set of at most K−1 arcs. It contrasts the polynomial-time solvability of packing out-branchings with the NP-completeness of deciding whether a general digraph contains arc-disjoint out-branching and in-branching structures.

The unresolved question asks whether sufficiently high arc-connectivity guarantees the existence of such an arc-disjoint pair, independently of computational complexity.