Kazashi and Sloan's number-theoretic conjecture for unshifted lattice rules

Let $n$ be an odd prime. For $1\leq\kappa\leq n-1$, define

$$T_n(\kappa):=\sum_{q=1}^{(n-1)/2}\frac{1}{q\,|r(q\kappa,n)|},$$

where $r(j,n)$ is the unique integer congruent to $j$ modulo $n$ in $\{-(n-1)/2,\ldots,(n-1)/2\}$. Kazashi and Sloan's conjecture. There exist constants $C_1,C_2>0$ and $\alpha\geq 2$, all independent of $n$, such that

$$T_n(\kappa)>C_1\frac{(\log n)^\alpha}{n}$$

holds for at most $C_2(\log n)^\alpha$ values of $\kappa$ among $\{1,\ldots,n-1\}$. The conjecture would improve the averaged worst-case error bound for unshifted rank-1 lattice rules to $O(1/n)$ up to a dimension-independent logarithmic factor; its resolution is not established in the supplied text.