Global Lipschitz continuity of the spherical cap discrepancy

Let $\mathbb{S}^{(d-1)}$ be the sphere and let $\Delta:\mathbb{S}^{(d-1)}\to [0,1]$ denote the spherical cap discrepancy. Global Lipschitz continuity conjecture. The spherical cap discrepancy $\Delta:\mathbb{S}^{(d-1)}\to [0,1]$ is Lipschitz continuous. The paper proves Lipschitz continuity around generic point sets and notes that local Lipschitz continuity around arbitrary point sets would imply the global result by compactness of the sphere. The global statement remains open in the source.