Nearly optimal spherical cap discrepancy for greedy sequences on the sphere

Let $\omega_{N,d} \subseteq \mathbb{S}^d$ be the first $N$ elements of the greedy sequence with respect to $K_{-1}$ as defined in the greedy construction. For $d$-dimensional spherical cap discrepancy, write $D_{L^2,\operatorname{cap}}$. Greedy discrepancy conjecture. For some $c=c(d)\geq 0$,

$$D_{L^2,\operatorname{cap}}(\omega_{N,d})=\mathcal{O}\bigl(N^{-\frac12-\frac{1}{2d}}\log^c(N)\bigr).$$

This conjecture collects the paper's numerical observations and predicts that the greedy sequence has nearly optimal spherical cap discrepancy. The supplied text does not state whether the bound is known or remains open.