The disjoint-transversal conjecture for row-Latin rectangles

Let $A$ be an $m\times n$ row-Latin rectangle based on $k$. A transversal selects one entry from each column, with all selected entries distinct, and pairwise disjoint transversals share no entries.

Disjoint-transversal conjecture. If $m\ge2n-1$, then $A$ has $m-(n-1)$ pairwise disjoint transversals.

This generalizes the observed behavior of the matrices $R_{m,n}$, for which the bound is attained. The claim is presented as an open problem for general row-Latin rectangles.