Linial–Luria discrepancy conjecture for Latin squares

Let $L$ be an $n\times n$ Latin square, viewed as the $n\times n\times n$ zero-one array $A(L)$ with $A_{i,x,q}=1$ if and only if $L_{i,x}=q$. A box is a set $T=I\times X\times Q$ with $I,X,Q\subseteq\{1,\ldots,n\}$, and its volume is

$$\operatorname{vol} T=|I||X||Q|.$$

Let $N_T(L)$ be the number of ones of $A(L)$ in $T$.

Linial–Luria's conjecture. There exist arbitrarily large Latin squares $L$ such that, for every box $T=I\times X\times Q$,

$$\mathopen{}\mathclose\bgroup\originalleft|N_T(L)-\frac{\operatorname{vol} T}{n}\aftergroup\egroup\originalright|=O\mathopen{}\mathclose\bgroup\originalleft(\sqrt{\operatorname{vol} T}\aftergroup\egroup\originalright).$$

This asks for Latin squares whose associated zero-one arrays have uniformly low discrepancy in every box, with the expected density $1/n$. The source presents it as a conjecture related to the quasirandomness of Latin squares; the paper proves that random Latin squares typically have relatively low discrepancy, but does not establish the asserted existence result.