The generalized McKay–Wanless conjecture for Latin rectangles

For $k\leq n$, let $\mathcal{LR}_{k,n}$ be the set of $k\times n$ Latin rectangles with symbol set $[n]$, and let $\randLR$ be chosen uniformly from this set. For every $\varepsilon>0$, the generalized McKay–Wanless conjecture. For all sufficiently large $n$ and $k\leq n$, (a) the expected number of order-$3$ Latin subsquares of $\randLR$ is

$$\left.(1\pm\varepsilon)\binom{k}{3}\middle/(3n^3)\right.$$

, and (b) $\randLR$ contains no Latin subsquare of order greater than $3$ with probability at least $1-\varepsilon$. This extends the corresponding conjecture for random Latin squares to Latin rectangles; the source presents it as a conjectural target while proving a weaker large-subsquare bound.