Rainbow matching conjecture in general graphs

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

Background

Conjecture 1.2 extends the bipartite rainbow matching conjecture to arbitrary graphs. As in the bipartite version, the matchings need not be edge-disjoint, and the conjecture asserts an exact n−1 lower bound for the size of a rainbow matching.

The paper reports only asymptotic progress toward this and related conjectures, so the exact statement remains explicitly identified as a conjecture.