Latin Rectangle Conjecture

For $\alpha,\beta\in\mathbb{N}$ with $\alpha\leq\beta$ and $\beta$ even, fix a set $\mathcal U$ of $\beta$ vectors in $\mathbb{C}^{\alpha}$ generically. A Latin Rectangle for $\mathcal U$ is an $\alpha\times\beta$ matrix whose entries are drawn from $\mathcal U$, with no entry occurring more than once in any row or column. For a Latin Rectangle $L$, let $\operatorname{coldet}(L)$ be the product over columns of the determinants formed from the column vectors. Latin Rectangle Conjecture. One has

$$\sum_L \operatorname{coldet}(L)\neq 0,$$

where the sum ranges over all Latin Rectangles for $\mathcal U$. This conjecture concerns the nonvanishing of an explicit sum of products of determinants for generic vectors; the source gives no evidence of resolution.