McKay–Wanless conjecture on subsquares in random Latin squares

Let $L$ be a random Latin square of order $n$, and let $cmathbb{E}_a(n)$ denote the expected number of subsquares of order $a$ in $L$. For $2\leqslant a\leqslant n/2$, McKay and Wanless's conjecture.

$$\mathbb{E}_a(n)= \begin{cases} \dfrac{n^2}{4}(1+o(1)) & \text{if } a=2,\\ \dfrac{1}{18}(1+o(1)) & \text{if } a=3,\\ o(1) & \text{if } 4\leqslant a\leqslant n/2. \end{cases}$$

The conjecture was a major goal in the study of subsquares of random Latin squares. Its $a=2$ asymptotic was established up to bounds by Kwan and Sudakov in 2018, and the remaining part referred to in the source was completely resolved by Kwan, Sah and Sawhney in 2022.