Brualdi–Stein conjecture on near transversals in Latin squares

A Latin square is an $n\times n$ array in which each symbol occurs exactly once in every row and column. A near transversal is a partial transversal of length $n-1$, meaning a set of $n-1$ entries chosen from distinct rows and columns with distinct symbols.

Brualdi–Stein conjecture. Every Latin square contains a near transversal.

This conjecture is attributed to Brualdi and Stein, and in another source to Ryser. It has been proved for Cayley tables of finite groups and for Latin squares of order at most $11$, but remains open in general.