Cycle chains or ladders in digraphs of large directed pathwidth

Prove or disprove whether every digraph of sufficiently large directed pathwidth contains a long cycle chain or a long directed ladder as a butterfly minor.

Background

The authors propose a more specific structural question that could provide insight into the broader problem of characterizing universal obstructions for directed pathwidth. The proposed obstruction families are the cycle chains and directed ladders already used in the paper for cycle rank.