  In this paper, we prove a quantitative spectral inequality for the second
Robin eigenvalue in non-compact rank-1 symmetric spaces. In particular, this
shows that for bounded domains in non-compact rank-1 symmetric spaces, the
geodesic ball maximises the second Robin eigenvalue among domains of the same
volume, with negative Robin parameter in the regime connecting the first
nontrivial Neumann and Steklov eigenvalues. This result generalises the work of
Freitas and Laugesen in the Euclidean setting [FL21] as well as our previous
work in the hyperbolic space [LWW20].
