Strong logarithmic discrepancy conjecture for arithmetic permutations

Let $p$ be prime, let $\psi_k$ be the arithmetic permutation considered in the paper, and let $D(\psi_k)$ denote its discrepancy. Strong logarithmic discrepancy conjecture.

$$\lim_{p \rightarrow \infty} \frac{\min_k D(\psi_k)}{\log p}=\frac12.$$

This is presented as a stronger form of the logarithmic discrepancy conjecture and would imply that, for every prime $p$ asymptotically, some $\psi_k$ is maximally quasirandom. The claim is motivated by a lower bound and computational evidence, but is not proved.