Strongly stable CMC-one hypersurfaces in every hyperbolic space of dimension at least four

For every integer $d\ge4$, we prove strong stability for a subfamily of classical rotational hypersurfaces in $\mathbb H^d$ with normalized mean curvature one. The examples are complete, two-sided, properly embedded, and nowhere umbilic, with topology $\mathbb {R}\times\mathbb{S}^{d-2}$. An explicit positive supersolution yields a quantitative stability inequality for all compactly supported test functions. Consequently, endpoint horospherical rigidity fails in every ambient dimension at least four.