The smooth $L_{\mathbf p}$-discrepancy lower-bound conjecture

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

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

The conjecture is supported by a theorem proving the same order under a bounded-weight condition. Its unrestricted weighted form remains open.