Strong bipartite rainbow matching conjecture

Prove that every collection of n matchings of size n+1 in a bipartite graph contains a rainbow matching of size n.

Background

Conjecture 1.3 asks for the minimum matching size that guarantees a full rainbow matching of size n when there are n matchings in a bipartite graph. Its threshold n+1 is asserted to be best possible, and the conjecture implies Conjecture 1.1 by adjoining a common dummy edge.

The paper states that this conjecture has been settled asymptotically, but it presents the exact assertion as an explicit conjecture.