Fundamental-solution conjecture for Helmholtz operators on spheres

Let $d\in\\{2,3,\ldots\\}$. Write $\mathbf S_R^d$ for the $d$-dimensional sphere of radius $R$, and let $\Delta$ denote its Laplace--Beltrami operator. The functions $\mathsf S_{R,\beta}^{d,-}(\mathbf x,\mathbf x^\prime)$ and $\mathsf A_{R,\beta}^{d,-}(\mathbf x,\mathbf x^\prime)$ are the candidate functions defined in the paper.

Fundamental-solution conjecture. The candidate function $\mathsf S_{R,\beta}^{d,-}(\mathbf x,\mathbf x^\prime)$ is a fundamental solution for the Helmholtz operator $(-\Delta-\beta^2)$ on $\mathbf S_R^d$. Similarly, the candidate function $\mathsf A_{R,\beta}^{d,-}(\mathbf x,\mathbf x^\prime)$ is an opposite antipodal fundamental solution for the Helmholtz operator $(-\Delta-\beta^2)$ on $\mathbf S_R^d$.

These assertions identify the candidate kernels with fundamental solutions on spheres in every integer dimension $d\geq 2$; the supplied text does not state whether they have been proved or remain open.