The disjoint-independent-transversal conjecture for matroid-valued matrices

Let $A$ be an $m\times n$ matrix whose entries belong to a matroid $M$. Suppose that the set of entries in each row forms an independent set of size $n$ in $M$. An independent transversal (IT) selects one entry from each column so that the selected entries form an independent set in $M$; pairwise disjoint ITs share no entries.

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

This generalizes the disjoint-transversal conjecture from row-Latin rectangles to matroid independence. The source presents it as an open generalization.