Neiderreiter's logarithmic discrepancy conjecture for arithmetic permutations

Let $p$) be prime, let $\mathbb{Z}_p^\times$ denote the nonzero residue classes modulo $p$, and let $\psi_k$ be the arithmetic permutation defined in the surrounding discussion. Write $D(\psi_k)$ for its discrepancy. Neiderreiter's conjecture. For some $k \in \mathbb{Z}_p^\times$, with $p$ prime, $D(\psi_k)=O(\log p)$. The best previously known bound is $O(\log p(\log\log p)^2)$, so the conjecture would give a substantially stronger logarithmic discrepancy estimate.