The harmonic-polynomial characterization of non-injectivity sets

Let $S\subset\mathbb R^d$. Call $S$ an injectivity set for the spherical mean transform if vanishing of all spherical means centered on $S$ forces the underlying function or distribution to vanish. Let $h$ be a harmonic homogeneous polynomial, let $a\in\mathbb R^d$, and let $V$ be an algebraic variety with $\dim V\le d-2$. The injectivity-set conjecture. If $S$ fails to be an injectivity set, then

$$S\subset (a+h^{-1}(0))\cup V.$$

In odd dimensions, the paper gives the equivalent formulation in terms of common nodal sets of Paley–Wiener families. The conjecture is established in the ruled-surface setting treated by the paper, while the general statement is not resolved there.