The conjecture on optimal lower bounds for weighted r-discrepancy

Let $D_q^r(\xi,\Lambda)$ be the $r$-discrepancy of a point set $\xi$ with weights $\Lambda$, and define

$$D_q^{r,o}(m,d):=\inf_{\xi,\Lambda}D_q^r(\xi,\Lambda).$$

Here $m$ is the number of points, $d$ is the dimension, and $C(r,d)>0$ depends only on $r$ and $d$. Weighted $r$-discrepancy conjecture. For all $d,r\in\mathbb N$,

$$D^{r,o}_\infty(m,d)\ge C(r,d)m^{-r}(\log m)^{d-1}.$$

The claim generalizes the conjectured optimal lower bound for ordinary discrepancy and is supported by results for even $r$ under a bound on the total variation of the weights; the unrestricted statement remains open.