Ryser–Brualdi–Stein conjecture on rainbow matchings

Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each part, and let its edges have a proper coloring with $n$ colors. A rainbow matching is a matching whose edges have pairwise distinct colors.

Ryser–Brualdi–Stein conjecture. Every such coloring contains a rainbow matching of size $n-1$; moreover, if $n$ is odd, it contains a perfect rainbow matching.

This is equivalent to the classical transversal problem for Latin squares: every Latin square has a partial transversal of size $n-1$, and odd-order Latin squares have a full transversal. The supplied text does not state a resolution.