The general finite-smoothness kernel Lagrange-basis conjecture

Let $\Xi$ be a quasiuniform center set with mesh ratio $\rho$, and let $\phi_{\beta}=G_{\beta}+G_{\beta}*\psi$ with $\beta>d$. Let $S(\phi_{\beta},\Xi)$ be the associated kernel space and $\chi_{\xi}$ its Lagrange functions. A local basis means the localization property in Assumption LagrangeDecay, and Assumption Bernstein is the stated Hölder-continuity condition. General kernel Lagrange-basis conjecture. If $\Xi$ is quasiuniform with mesh ratio $\rho$, then the Lagrange functions $\chi_{\xi}\in S(\phi_{\beta},\Xi)$ form a local basis and satisfy Assumption Bernstein. This is posed as a more ambitious extension covering kernels of the displayed form, including compactly supported surface-basis functions and many finite-smoothness kernels; the source states that the consequences listed for the spherical case would then follow, but gives no resolution.