The rainbow matching conjecture for general graphs

Let $(m,n)\to k$ mean that every $m$ matchings of size $n$ in any graph have a rainbow matching of size $k$. The notation $(m,n)\to_{\mathcal{B}} k$ has the same meaning for bipartite graphs. Rainbow matching conjecture.

$$(2n,n)\to n.$$

If $n$ is odd, then

$$(2n-1,n)\to n.$$

The bipartite analogue $(2n-1,n)\to_{\mathcal{B}}n$ is known by Drisko's theorem. The conjecture asserts that passing from bipartite graphs to general graphs costs only one additional matching; it remains open in the supplied source.