The conjectural optimal supremum-norm discrepancy bound

Let $D_N$ be the discrepancy function of an $N$-point set $\mathcal P_N\subset[0,1]^d$. Supremum-norm discrepancy conjecture. In dimensions $d\ge 3$ there holds

$$\lVert D_N\rVert_{\infty}\gtrsim(\log N)^{d/2}.$$

This would improve the known lower bound by an additional logarithmic power; the paper presents it as the expected optimal form of the $L^\infty$ estimate.