Stein's conjecture on transversals in Latin arrays

An $m\times n$ array is row-Latin if each row contains each symbol at most once, and column-Latin if each column does so; it is Latin when it is both row-Latin and column-Latin. A transversal in an $(n-1)\times n$ array is a set of $n-1$ entries chosen from distinct rows and columns with distinct symbols.

Stein's conjecture. Let $R$ be an $(n-1)\times n$ Latin array. Then $R$ contains a transversal.

This is the only one of Stein's seven listed conjectures that remains unsolved according to the source. The stronger formulation requiring only a row-Latin array is therefore also presented as the motivating open problem for the Latin case.