The surface-spline Lagrange-basis conjecture on $SO(3)$

Let $\Xi$ be a quasiuniform subset of $SO(3)$ with mesh ratio $\rho$. For the kernel $k_m$ on $SO(3)$ and its associated space $S(k_m,\Xi)$, let $\chi_{\xi}$ denote the Lagrange function associated with $\xi\in\Xi$. A local basis means the localization property in Assumption LagrangeDecay, and Assumption Bernstein is the stated Hölder-continuity condition. $SO(3)$ surface-spline conjecture. If $\Xi\subset SO(3)$ is quasiuniform with mesh ratio $\rho$, then the Lagrange functions $\chi_{\xi}\in S(k_m,\Xi)$ form a local basis and satisfy Assumption Bernstein. The conjecture is the $SO(3)$ analogue of the preceding kernel-basis conjectures; the source notes that these kernels have theoretical precise error estimates, but does not state whether this conjecture is resolved.