The spherical restricted-surface-spline Lagrange-basis conjecture

Let $\Xi$ be a quasiuniform set of centers on the sphere, with mesh ratio $\rho$. For restricted surface splines $\varphi_m$ and the associated conditionally positive-definite kernel space $S(\varphi_m,\Xi)$, let $\chi_{\xi}$ denote the Lagrange function associated with $\xi\in\Xi$. A local basis means the localization property in Assumption LagrangeDecay, and Hölder continuity means the regularity condition in Assumption Bernstein. Spherical restricted-surface-spline conjecture. If $\Xi$ is quasiuniform with mesh ratio $\rho$, then the Lagrange functions $\chi_{\xi}\in S(\varphi_m,\Xi)$ form a local basis and satisfy the Hölder continuity assumption. If true, the conjecture would yield bounded Lebesgue constants, precise interpolation rates, stability of the Lagrange basis, bounded $L_p$ norms for the associated $L_2$ projector, and corresponding precise $L_p$ approximation rates, as described in the source. The source does not specify whether the conjecture has been resolved.