Higher-dimensional characterization of non-uniqueness sets for the spherical mean transform

Let $d$ be the ambient dimension and let $S\subset\mathbb{R}^d$. A set $S$ is a non-uniqueness set for the spherical mean transform in the space of compactly supported functions when there exists a nonzero homogeneous harmonic polynomial whose zero surface is denoted by $\Sigma$, a rigid motion $\omega$ of $\mathbb{R}^d$, and an algebraic surface $\Phi$ of dimension at most $d-2$. Higher-dimensional non-uniqueness conjecture. A set $S\subset\mathbb{R}^d$ is a non-uniqueness set if and only if

$$S\subset\omega\Sigma\cup\Phi.$$

This conjecture proposes the higher-dimensional analogue of the known planar characterization, in which non-uniqueness sets are contained in a rigid motion of a Coxeter system together with a finite set. The corresponding characterization in higher dimensions is presented as an open problem.