Sharp shifted reciprocal sums of Neumann eigenvalues on space forms

Let $\mathbb{M}_\kappa^n$ be the space form of sectional curvature $\kappa\in\{-1,0,1\}$, so that $\mathbb{M}_{-1}^n=\mathbb{H}^n, \mathbb{M}_{0}^n=\mathbb{R}^n, \mathbb{M}_{1}^n=\mathbb{S}^n$. Let $\Omega\subset\mathbb{M}_\kappa^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<|\Omega|<|\mathbb{S}^n|$ when $\kappa=1$. Write $0=\mu_0(\Omega)\leq\mu_1(\Omega)\leq\cdots$ for the Neumann spectrum, and let $B_R^\kappa\subset\mathbb{M}_\kappa^n$ be a geodesic ball of volume $\vert\Omega\vert/2$. We prove the sharp shifted reciprocal inequality $$\sum_{j=2}^{n+1}\frac1{\mu_j(\Omega)} \geq \frac{n}{\mu_1(B_R^\kappa)} = \frac{n}{\mu_2(B_R^\kappa\sqcup B_R^\kappa)}.$$ Equality holds if and only if $\Omega$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.