The discrepancy-maximization conjecture for the middle slice

Let $A\in\binom{[n]}{n/2}$, and define \textit{Disc-\max-}d(A)=1 if and only if

$$\left|\lvert A\cap [j]\rvert-j/2\right|\le d/2\quad\text{for all }1\le j\le n.$$

Equivalently, if $disc(j)$ is the signed discrepancy of the prefix $[j]$, then \textit{Disc-\max-}d(A) is true exactly when $|disc(j)|\le d$ for every $1\le j\le n$. Discrepancy-maximization conjecture. For a suitable constant $d$,

E_{n/2}(\textit{Disc-\max-}d)=O(1).

This proposes an explicit discrepancy-based Boolean function whose query complexity on the middle slice has bounded excess. The paper presents it as a candidate and does not establish the claim.