Approximate independence for sparse partial Latin rectangles

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$. A partial Latin rectangle is $\beta n$-sparse if each row and column contains at most $\beta n$ nonempty entries and each symbol is used at most $\beta n$ times. If $P$ has $\ell$ nonempty entries and $\randLR$ is chosen uniformly from $\mathcal{LR}_{k,n}$, the sparse partial-rectangle conjecture. For every $\varepsilon>0$, there exists $\beta>0$ such that, for all sufficiently large $n$ and all $k\leq n$,

$$\left(\frac{1-\varepsilon}{n}\right)^\ell\leq\Pr(P\subseteq\randLR)\leq\left(\frac{1+\varepsilon}{n}\right)^\ell.$$

This formalizes the paper's probabilistic heuristic that sufficiently sparse compatible entries of a random Latin rectangle behave approximately independently; the paper proves the main sparse-rectangle probability estimate in a restricted range, so the stated conjecture is presented as a broader target.