Drisko's uniqueness conjecture for transversal-free row-Latin rectangles

Let $n\ge2$, and let $A$ be a $(2n-2)\times n$ row-Latin rectangle based on $k$. A transversal selects one entry from each column, with all selected entries distinct. The matrix $R_{2n-2,n}$ is the standard example of a row-Latin rectangle without a transversal.

Drisko's conjecture. Either $A$ has a transversal, or $A$ can be transformed into $R_{2n-2,n}$ by permuting rows, columns, and symbols.

This asserts that, up to the stated equivalences, $R_{2n-2,n}$ is the unique transversal-free matrix of this size. The source records that the corresponding uniqueness statements for its other conjectures are false, but does not state a resolution of Drisko's conjecture here.