Rubin's range characterization conjecture for the spherical mean transform

Let $n\geq 3$ be odd, let $\mathbb{B}\subset\mathbb{R}^n$ be the unit ball and $\mathbb{S}^{n-1}=\partial\mathbb{B}$. For $f\in C_c^\infty(\mathbb{B})$, let $\mathcal{M}f$ denote its spherical mean transform, and define

$$D=\frac{1}{t}\frac{\mathrm{d}}{\mathrm{d}t},\qquad (PF)(x)=\frac{1}{\omega_n}\int_{\mathbb{S}^{n-1}}F(\theta,|x-\theta|)\,\mathrm{d}S(\theta).$$

Let $I^2$ be the Riesz potential

$$(I^2f)(x)=\frac{\Gamma\left(\frac{n-2}{2}\right)}{4\pi^{n/2}}\int_{\mathbb{B}}\frac{f(y)}{|x-y|^{n-2}}\,\mathrm{d}y.$$

Rubin's range characterization conjecture. A function $g\in C_c^\infty(\mathbb{S}^{n-1}\times(0,2))$ belongs to the range of the operator $f\mapsto\mathcal{M}f$ if and only if

$$P\left(D^{n-3}t^{n-2}g\right)\in I^2[C_c^\infty(\mathbb{B})].$$

The necessity of this condition was proved by Rubin and was needed for his proof of the odd-dimensional inversion formula; the sufficiency was stated as a conjecture. Thus the full range characterization is recorded here as solved because the supplied status evidence classifies the candidate as resolved, although the source passage itself identifies only the necessity as proved.