Chow–Fan–Goemans–Vondrák wideness conjecture for Latin tableaux

Let $\lambda$ be a partition, and let a Latin tableau of shape $\lambda$ be a filling of the Young diagram of shape $\lambda$ such that row $i$ contains the numbers in $[\lambda_i]$ in some order and no number appears more than once in a column. A Young diagram $\lambda$ is called wide when it satisfies the wideness condition defined in the paper.

Chow–Fan–Goemans–Vondrák conjecture. A Young diagram $\lambda$ is wide if and only if there exists a Latin tableau of shape $\lambda$.

Latin tableaux generalize Latin squares and rectangles to arbitrary partition shapes. Wideness is necessary for the existence of a Latin tableau, while sufficiency remains open.