Strong general-graph rainbow matching conjecture

Prove that every collection of n matchings of size n+2 in a graph contains a rainbow matching of size n.

Background

Conjecture 1.4 is the non-bipartite analogue of Conjecture 1.3. It specifies the conjectured threshold n+2 for guaranteeing a rainbow matching of size n among n matchings in an arbitrary graph.

The paper gives an extremal construction showing that the threshold cannot generally be reduced and notes that the conjecture has only been settled asymptotically.