The tubular geometric control conjecture for $L^p$-Logvinenko-Sereda sets on spheres

Let $S^m$ be the standard sphere, let $1\le p<\frac{2m}{m-1}$, and let $\mathcal{A}=\{A_\lambda\}_{\lambda\ge1}$ be a family of subsets of $S^m$. An $L^p$-Logvinenko-Sereda family for eigenfunctions means that the corresponding eigenfunctions satisfy a uniform $L^p$ sampling inequality on the sets $A_\lambda$. The tubular geometric control condition is the condition referred to in the source for this family. Tubular geometric control conjecture. $\mathcal{A}$ is $L^p$-Logvinenko-Sereda for eigenfunctions on the standard sphere $S^m$ if and only if $\mathcal{A}$ satisfies the tubular geometric control condition. This extends the known characterization for $p>\frac{2m}{m-1}$ to the complementary range. The conjecture is motivated by examples showing that relative density and symmetric relative density are not necessary for small $p$, together with evidence from concentrated spherical harmonics and control theory; the source presents it as open.