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 $\mathrm{\Omega }\subset {\mathbb{M}}_{\kappa }^n$ be a nonempty bounded open set with Lipschitz boundary, and assume that $0<\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }<\mathrm{\mid }{\mathbb{S}}^n\mathrm{\mid }$ when $\kappa =1$. Write $0={\mu }_0(\mathrm{\Omega })\le {\mu }_1(\mathrm{\Omega })\le \cdots$ for the Neumann spectrum, and let $B_R^{\kappa }\subset {\mathbb{M}}_{\kappa }^n$ be a geodesic ball of volume $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}2$. We prove the sharp shifted reciprocal inequality $$\sum _{j=2}^{n+1}\frac{1}{{\mu }_j(\mathrm{\Omega })}\ge \frac{n}{{\mu }_1(B_R^{\kappa })}=\frac{n}{{\mu }_2(B_R^{\kappa }\bigsqcup B_R^{\kappa })}\mathrm{.}$$ Equality holds if and only if $\mathrm{\Omega }$ is the disjoint union of two equal geodesic balls. This gives an affirmative answer to the conjecture of \cite[Remark~11]{BucurMartinetNahon2025}.