The optimal-order discrepancy conjecture for finite point sets

Let $D(m,d)_\infty:=\inf_T D(T,m,d)_\infty$ denote the minimum, over finite point sets $T$ with $m$ points in dimension $d$, of their star discrepancy. For $d\geq 3$, the conjectured lower bound is

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

Optimal-order discrepancy conjecture. For every $d\geq 3$, there is a constant $C(d)>0$ such that

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

The matching upper bound $D(m,d)_\infty\leq C(d)m^{-1}(\log m)^{d-1}$ is known, and the lower bound is known when $d=2$ by Schmidt's result. Establishing it for dimensions $d\geq 3$ is described as a major open problem in discrepancy theory.