Edmonds–Giles conjecture for 5-chordal underlying graphs

Prove that the Edmonds–Giles conjecture holds for every digraph whose underlying undirected graph is 5-chordal, equivalently, whose underlying graph has no chordless cycle of length greater than 5.

Background

The paper proves the Edmonds–Giles conjecture for digraphs whose underlying undirected graphs are chordal, meaning that they contain no chordless cycle longer than a triangle. It then proposes extending this result to 5-chordal graphs. Because transitively closed digraphs reduce odd chordality by one, the authors observe that it would suffice to establish the result for 4-chordal graphs; the length-six chordless cycle in Schrijver’s counterexample indicates why this proposed extension is the strongest possible within the stated framework.