The smooth discrepancy lower-bound conjecture

For a point set $\xi$ with $m$ points in the $d$-dimensional domain and arbitrary weights, let $D^{r,o}_\infty(m,d)$ be the infimum of the $r$-discrepancy over all such point-weight pairs. Smooth discrepancy lower-bound conjecture. For all $d,r\in\mathbb N$,

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

This generalizes the classical discrepancy lower-bound conjecture to smooth discrepancy and weighted formulas. The stated bound is supported by known lower estimates under restrictions on the weights, but the unrestricted assertion remains open.