The Wide Partition Conjecture for free matroids

Let $\lambda$ be an integer partition. A wide partition is one for which every subpartition $\mu\subseteq\lambda$ dominates its conjugate $\mu'$; equivalently, $\mu\geq\mu'$ in dominance order for every subpartition $\mu$. A tableau of shape $\lambda$ is a filling of its Young diagram with integers.

The Wide Partition Conjecture for free matroids. An integer partition $\lambda$ is wide if and only if there exists a tableau of shape $\lambda$ such that (1) for every $i$, the entries in the $i$th row are precisely the integers from $1$ to $\lambda_i$ inclusive, and (2) for every $j$, the entries in the $j$th column are pairwise distinct.

This is the free-matroid case of the broader Wide Partition Conjecture, which is connected to Rota's basis conjecture. The claim had been verified by computer for partitions with at most 65 cells and for partitions whose diagrams fit inside a $10\times10$ square, but no general proof is given here.