Universality conjecture for positive-discrepancy entry distributions

Let $\mu$ be a probability measure with mean $0$ and variance $1$, and define its discrepancy by

$$\Delta(\mu)=\inf_{n\geq 1}\inf_{x_1,\ldots,x_n\in\operatorname{supp}(\mu),\ s_1,\ldots,s_n\in\{\pm1\}}\left|\sum_{i=1}^n s_i x_i\right|.$$

Let $\beta(\mu,\infty)$ denote the scaling exponent of the runtime of $\infty$-reluctant dynamics. Positive-discrepancy universality conjecture. $\beta(\mu,\infty)\approx 1.6$ is a constant independent of $\mu$ for any “sufficiently nice” $\mu$ with mean $0$, variance $1$, and discrepancy $\Delta(\mu)>0$. This identifies positive-discrepancy distributions as a proposed universality class for the runtime exponent, but the claim remains conjectural.