Positive topological ordering, take 3
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.