The Brualdi–Ryser conjecture for partial transversals in latin squares

Let $L$ be a latin square of order $n$. A near transversal is a partial transversal of size $n-1$, and a transversal is a partial transversal of size $n$.

Brualdi–Ryser conjecture. Every latin square $L$ of order $n$ possesses a near transversal, and if $n$ is odd then $L$ possesses a transversal.

These are longstanding conjectures on the existence of large partial transversals in latin squares. The source explains that the chromatic number conjecture would imply them; their status remains open.