The weighted higher-order discrepancy lower-bound conjecture

Let $D_r^w(m,d)_\infty$ denote the minimal weighted $r$-discrepancy in dimension $d$, where $m$ is the number of nodes. Weighted higher-order discrepancy lower-bound conjecture. For all $d,r\in\mathbb N$,

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

This generalizes the conjectured lower bound for the classical star-discrepancy; the source presents it as an open problem.