Brualdi–Ryser–Stein conjecture for partial transversals in Latin squares

Let $L$ be an $n\times n$ Latin square. A partial transversal is a set of entries with no two in the same row, column, or using the same symbol.

Brualdi–Ryser–Stein conjecture. Every $n\times n$ Latin square has a partial transversal of size $n-1$.

This is equivalent to the existence of a rainbow matching of size $n-1$ in every properly edge-colored $K_{n,n}$ using $n$ colors. The conjecture remains open; the best results give asymptotic partial transversals of size $n-o(n)$.