Grid minor in digraphs 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And I'm looking for this unpublished paper (how they proved the conjecture for di-planar graphs), or related stuff in this case, actually how to use such a grid (I mean $J_k$).
There is a new preprint by Stephan Kreutzer and Ken-ichi Kawarabayashi, in which they apparently show that the statement (5.1) is true for all digraphs.

Stephan Kreutzer and Ken-ichi Kawarabayashi: The directed grid theorem.    arXiv:1411.5681 [cs.DM]  

EDIT (June 16, 2015):

A short version of their paper appears here:

Ken-ichi Kawarabayashi, Stephan Kreutzer. The Directed Grid Theorem. In: Rocco A. Servedio, Ronitt Rubinfeld (eds.), Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing 2015. pp. 655-664