Quasi-Monte Carlo convergence conjecture for mapped $(0,2)$-sequences on the sphere

Let $\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots \in [0,1)^2$ be a $(0,2)$-sequence, and let $\boldsymbol{z}_0,\boldsymbol{z}_1,\ldots \in \mathbb{S}^2$ be the corresponding points on the sphere obtained using the mapping $\Phi$. For $N\geq 1$, define the quadrature rule

$$Q_N(f)=\frac{1}{N}\sum_{n=0}^{N-1}f(\boldsymbol{z}_n).$$

Convergence conjecture. The squared worst-case error in the Sobolev space $\mathcal{H}^{3/2}$ satisfies

$$e^2(Q_N,\mathcal{H}^{3/2})=\mathcal{O}(N^{-3/2})\qquad\text{as }N\to\infty.$$

This would establish the expected quasi-Monte Carlo convergence rate for digital $(0,2)$-sequences mapped from the unit square to $\mathbb{S}^2$. The claim is motivated by the numerical results presented in the paper; no proof or resolution is supplied in the source.