Woźniakowski's conjecture on the exponent of tractability of star discrepancy

Let

$$\overline{\alpha}:=\sup\left\{\alpha\mid \exists c,k\geq 0\ \forall N,d\in\mathbb{N}\ \exists P\subset[0,1]^d:\ |P|=N\ \wedge\ D_N^*(P)\leq c\frac{d^k}{N^\alpha}\right\}.$$

Here $D_N^*(P)$ denotes the star discrepancy of an $N$-point set $P$.

Woźniakowski's conjecture.

$$\overline{\alpha}=\frac{1}{2}.$$

This conjecture concerns the optimal dependence on the number of points in high-dimensional star discrepancy bounds. The paper states that the known bounds give $1/2\leq\overline{\alpha}<0.9373$, and that the problem remains unsolved.