High-dimensional permutation discrepancy conjecture

Let $d$ be a positive integer. A $d$-dimensional permutation is an $[n]^{d+1}$ array of zeros and ones with exactly one $1$ on every line parallel to any coordinate axis. For a box $\mathcal T=T_1\times\cdots\times T_{d+1}\subseteq[n]^{d+1}$, define its volume and the number of permutation entries it contains by

$$\operatorname{vol}(\mathcal T)=\prod_i|T_i|,\qquad A(\mathcal T)=|\{\alpha\in\mathcal T:A(\alpha)=1\}|.$$

High-dimensional permutation discrepancy conjecture. For every $d\geq2$ there exist arbitrarily large $d$-dimensional permutations $A$ such that, for every box $\mathcal T$,

$$\left|A(\mathcal T)-\frac{\operatorname{vol}(\mathcal T)}{n}\right|=O(\sqrt{\operatorname{vol}(\mathcal T)}).$$

This is a high-dimensional analogue of the expander mixing lemma and predicts optimal square-root discrepancy for all boxes. The paper gives heuristic motivation but no resolution of the conjecture.