McKay–Wanless conjecture on intercalates in random Latin squares

Let $\mathcal{L}$ be the set of $n\times n$ Latin squares, and let $\boldsymbol{L}\in\mathcal{L}$ be uniformly random. Let $\boldsymbol{N}=N(\boldsymbol{L})$ denote the number of intercalates in $\boldsymbol{L}$. For any fixed $\varepsilon>0$, asymptotically almost surely,

$$\mathopen{}\mathclose\bgroup\originalleft(1-\varepsilon\aftergroup\egroup\originalright)\frac{n^{2}}{4}\le\boldsymbol{N}\le\mathopen{}\mathclose\bgroup\originalleft(1+\varepsilon\aftergroup\egroup\originalright)\frac{n^{2}}{4}.$$

McKay–Wanless conjecture. The displayed concentration holds for every fixed $\varepsilon>0$. This predicts that a uniformly random Latin square has approximately $n^{2}/4$ intercalates. The paper proves a matching lower bound asymptotically almost surely and obtains an upper bound of $fn^{2}$ for every function $f\to\infty$, but the sharp asymptotic upper bound remains open.