Agranovsky–Quinto conjecture on higher-dimensional non-injectivity sets

Let $S\subset\mathbb{R}^n$ and let $C_c(\mathbb{R}^n)$ denote the compactly supported continuous functions on $\mathbb{R}^n$. A rigid motion of $\mathbb{R}^n$ is denoted by $\omega$, let $\Sigma$ be the zero set of a homogeneous harmonic polynomial, and let $F\subset\mathbb{R}^n$ be an algebraic subset of co-dimension at least $2$. Agranovsky–Quinto conjecture. The following condition is necessary and sufficient for $S$ to be a set of injectivity for the circular Radon transform on $C_c(\mathbb{R}^n)$: $S$ is not contained in any set of the form

$$\omega(\Sigma)\bigcup F.$$

This conjecture extends the known two-dimensional characterization of injectivity sets, where the exceptional sets are rigid motions of Coxeter systems together with finite sets. It describes the expected non-injectivity sets in higher dimensions; the supplied source does not state a resolution.