Ryser–Brualdi–Stein conjecture on Latin square transversals

Prove that every n × n Latin square has a transversal of size n − 1, and furthermore a full transversal when n is odd.

Background

Montgomery proved the existence of a transversal of size n−1 for all sufficiently large n, settling the first part asymptotically. The full-transversal existence when n is odd remains open in general and is the central remaining part of the combined Ryser–Brualdi–Stein conjecture.