The greedy-Haar thinning discrepancy conjecture

Let $d\in\mathbb{N}$ and let $\beta\in(0,1]$. A greedy-Haar $(1+\beta)$-thinning strategy produces a sequence $Z=(Z_n)_{n\geq 1}$, and write $Z^n=(Z_1,\ldots,Z_n)$ for its first $n$ points. Let $\operatorname{Dis}(Z^n)$ denote their discrepancy. Greedy-Haar thinning discrepancy conjecture. The sequence $Z$ almost surely satisfies

$$\limsup_{n\to\infty}\frac{\operatorname{Dis}(Z^n)}{\log^{d+1}(n)}<\infty.$$

This conjecture predicts a further logarithmic improvement for the empirically more efficient greedy-Haar strategy, beyond the conjectured bound for the Haar strategy; it is presented as a heuristic conjecture and remains open in the source.