Almost-everywhere optimal empty-box conjecture for Latin squares

For a $d$-dimensional permutation $A$, let $\mathcal T\subseteq[n]^{d+1}$ be a box and write $A(\mathcal T)$ for the number of $1$-entries of $A$ in it. An empty box is a box with $A(\mathcal T)=0$, and let $\varepsilon(A)$ be the maximum volume of an empty box. In particular, for an order-$n$ Latin square, $\varepsilon(A)$ is the maximum volume of an empty box in its associated three-dimensional permutation. Almost-everywhere optimal empty-box conjecture. Asymptotically almost every order-$n$ Latin square $A$ satisfies

$$\varepsilon(A)=O(n^2).$$

The conjecture would improve the paper's bound $O(n^2\log^2 n)$ for almost every Latin square to the optimal order, matching the universal lower bound $\varepsilon(A)=\Omega(n^2)$ and the infinitely many examples with $\varepsilon(A)=O(n^2)$.