The higher-dimensional characterization of non-uniqueness sets in photoacoustic tomography

Let $n$ be a positive integer, let $S\subset\mathbb{R}^n$, and let $f$ be compactly supported. A set $S$ is a non-uniqueness set for compactly supported functions $f$ if the corresponding acquisition data do not uniquely determine $f$. Let $\Sigma$ be the cone of zeros of a homogeneous harmonic polynomial, homogeneous with respect to some point in $\mathbb{R}^n$, and let $\Phi$ be an algebraic subset of $\mathbb{R}^n$ of dimension at most $n-2$. The higher-dimensional non-uniqueness-set conjecture. The set $S$ is a non-uniqueness set for compactly supported functions $f$ if and only if

$$S\subseteq\Sigma\cup\Phi.$$

This conjectures the $n$-dimensional analogue of the known planar characterization of non-uniqueness sets, where, up to finitely many points, the set is a Coxeter cross. The result is stated as unproved in the source; only partial advances toward the higher-dimensional case are known.