Non-universality conjecture for zero-discrepancy entry distributions

Let $\mu_1,\mu_2,\mu_3$ be probability measures with mean $0$ and variance $1$, and let $\Delta(\mu)$ be the discrepancy of a probability measure. Let $\beta(\mu,\infty)$ denote the runtime scaling exponent of $\infty$-reluctant dynamics. Zero-discrepancy non-universality conjecture. There exist $\mu_1,\mu_2,\mu_3$ such that $\Delta(\mu_1)>0$, $\Delta(\mu_2)=\Delta(\mu_3)=0$, and the three numbers $\beta(\mu_1,\infty)$, $\beta(\mu_2,\infty)$, and $\beta(\mu_3,\infty)$ are all distinct. This predicts that positive-discrepancy distributions and zero-discrepancy distributions do not share one universal runtime exponent, while the conjecture remains open in the supplied text.