The $L^p$ discrepancy conjecture below the endpoint

Let $\mathcal A_N\subset[0,1]^d$ be a set of cardinality $N$, and let $D_N$ be its discrepancy function,

$$D_N(\vec x)=\sharp\bigl(\mathcal A_N\cap[0,\vec x)\bigr)-N\lvert[0,\vec x)\rvert.$$

For $0<p<1$, write $\lVert D_N\rVert_p$ for its $L^p$ quasi-norm.

The subunit-$L^p$ discrepancy conjecture. For every dimension $d\geq2$ and every $0<p<1$, there is a constant depending only on $d$ and $p$ such that

$$\lVert D_N\rVert_p\gtrsim(\log N)^{(d-1)/2}.$$

This is presented as an extension of the $L^1$ conjecture. The source states that nothing was known about it, even in dimension two, so it remains open.