Thor Johnson, et al, in their paper: Directed Tree Width, introduced a definition for directed grid $J_k$, and they conjectured:
$(5.1)$ For every integer $k$ there exists an integer $N$ such that every digraph with tree-width $N$ or more has a minor isomorphic to $J_k$.
And they continued by saying:
We have convinced ourselves that $(5.1)$ holds for planar digraphs, but the general case is open.
And I'm looking for this unpublished paper (how they proved the conjecture for di-planar graphs), or related stuff in this case, actually how to use such a grid (I mean $J_k$).