Positive topological ordering, take 3 Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix?

This question is motivated by this problem:
Positive topological ordering

The original decision problem is at least as hard as this one, so an NP-completeness result would solve that too.

Edit:
Laszlo Vegh and Andras Frank called my attention to an equivalent problem asked by Gunter Rote:
http://lemon.cs.elte.hu/egres/open/Graphs_extendable_to_a_uniquely_matchable_bipartite_graph

Edit:
The reduction to the original problem is as follows. Suppose that the DAG has only two levels, these will correspond to the rows and columns of the matrix. Also, we have one single node with weight +1. Everyone else in the lower level has weight -1 and on the upper level +1.
The problem turned out to be NP-complete. You can read more in detail here and here. Short summary:

The reduction is from a problem was shown to be NP-complete by Dasgupta, Jiang, Kannan, Li, and Sweedyk: given a bipartite graph G and an integer k, decide if G has an induced subgraph on 2k nodes that can be extended to be uniquely matchable. It was observed by Stéphane Vialette that this reduces to the bipartite unique matching version of this problem if we add n-k isolated nodes to both classes.