Rainbow paths from minimum out-degree

Determine whether every properly edge-coloured digraph with minimum out-degree d contains a rainbow path of length d-o(d).

Background

The main path results assume regularity or near-Eulerian degree conditions, whereas a simple greedy argument gives only a rainbow path of length at least d/2 from a minimum out-degree assumption.

The authors explicitly state that they know no improvement to this bound for directed graphs and ask whether the asymptotically optimal length can nevertheless be guaranteed.