Berger–Aharoni conjecture on rainbow matchings

Let $G$ be a bipartite graph, and let $M_1,\ldots,M_n$ be matchings in $G$, each of size $n$. A partial rainbow matching is a matching containing at most one edge from each $M_i$. Berger–Aharoni conjecture. The matchings $M_1,\ldots,M_n$ have a partial rainbow matching of size $n-1$. This conjecture is closely related to open problems on transversals of Latin squares, including conjectures of Ryser, Brualdi, and Stein.