Reciprocal sums of Neumann eigenvalues in non-Euclidean space forms

Let $M_{\kappa }^n$ be the simply connected space form of dimension $n\ge 2$ and constant sectional curvature $\kappa \in \{-1,1\}$. For every bounded connected smooth domain $\mathrm{\Omega }\subset M_{\kappa }^n$, assume in the case $\kappa =1$ that $\mathrm{\Omega }$ is contained in an open hemisphere, and let $B_{\mathrm{\Omega }}$ be a geodesic ball with $\mathrm{\mid }{B}_{\mathrm{\Omega }}\mathrm{\mid }=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$. We prove $$\sum _{j=1}^n\frac{1}{{\mu }_j(\mathrm{\Omega })}\ge \frac{n}{{\mu }_1(B_{\mathrm{\Omega }})},$$ where ${\mu }_j(\mathrm{\Omega })$ are the positive Neumann eigenvalues of $\mathrm{\Omega }$. Equality holds if and only if $\mathrm{\Omega }$ is a geodesic ball. This proves a conjecture proposed by Xia and Wang [Math. Ann. 385, 2023, 863-879].