Chow et al.'s Latin Young diagram conjecture

Let $Y$ be a Young diagram with row lengths $l_1,\ldots,l_m$. It is Latin if its cells can be assigned integers so that row $i$ contains $1,\ldots,l_i$, and the entries in each column are distinct. A Young diagram is wide if every subdiagram formed by a subset of its rows dominates its conjugate. Chow et al.'s conjecture. Every wide Young diagram is Latin. Chow et al. proved this in some special cases, and the paper proves it for Young diagrams with three distinct row lengths; the assertion remains open in general.