Reciprocal sums of Neumann eigenvalues in non-Euclidean space forms

Let $M^n_\kappa$ be the simply connected space form of dimension $n\ge2$ and constant sectional curvature $\kappa\in\{-1,1\}$. For every bounded connected smooth domain $\Omega\subset M^n_\kappa$, assume in the case $\kappa=1$ that $\Omega$ is contained in an open hemisphere, and let $B_\Omega$ be a geodesic ball with $|B_\Omega|=|\Omega|$. We prove $$
\sum_{j=1}^n \frac1{\mu_j(\Omega)}\ge \frac{n}{\mu_1(B_\Omega)}, $$ where $\mu_j(\Omega)$ are the positive Neumann eigenvalues of $\Omega$. Equality holds if and only if $\Omega$ is a geodesic ball. This proves a conjecture proposed by Xia and Wang [Math. Ann. 385, 2023, 863-879].