Avoiding the directed grid theorem in the cycle-rank obstruction theorem

Determine whether the universal butterfly-minor obstruction theorem for digraphs of large cycle rank can be proved without using the directed grid theorem, potentially yielding a significantly improved bound for the obstruction function.

Background

The main theorem derives unavoidable directed ladders, directed cycle chains, or directed tree chains as butterfly minors from sufficiently large cycle rank. Its quantitative bound depends on the function supplied by the directed grid theorem. The authors explicitly ask whether the structural result can be obtained without that theorem, because avoiding the grid theorem could substantially improve the bound.