Rainbow matching conjecture in bipartite graphs

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

Background

Conjecture 1.1 generalizes the Ryser–Brualdi–Stein setting by allowing the n matchings to overlap and by not requiring them to cover the same vertex set. The conjecture asks for a rainbow matching using n−1 distinct colours from arbitrary matchings of size n in a bipartite graph.

The paper states that this conjecture has been settled only asymptotically, with known results guaranteeing a rainbow matching of size n−o(n), rather than the exact n−1 bound.