Directed paths of full average-degree length in digon-free Eulerian digraphs

Determine whether every Eulerian digraph with no digons contains a directed path whose length is at least the underlying average degree of the digraph.

Background

The paper proves that an Eulerian digraph contains a directed path of length at least half its underlying average degree, using a permutation-prefix argument. It then notes that the argument can be improved to obtain a factor of $\log 2$ when the digraph contains no digons (that is, no pair of opposite arcs).

The unresolved issue is whether the factor can be increased to one: namely, whether the length of a longest directed path must be at least the underlying average degree. This question is later formulated as a conjecture for Eulerian oriented graphs, but the earlier passage gives the clearest explicit open-problem statement.