  It has been shown that the synergy of a scalar field coupling with both the
Ricci scalar and the Gauss-Bonnet invariant significantly affects the
properties of scalarized black holes and neutron stars, including their domain
of existence and the amount of scalar hair they carry. Here we study the radial
stability of scalarized black-hole solutions. We demonstrate that they are
stable against radial perturbations for Ricci couplings consistent with both a
late-time cosmological attractor and the evasion of binary pulsar constraints.
In addition, we investigate the effect of the Ricci coupling on the
hyperbolicity of the equation governing linear, radial perturbations and show
that it significantly reduces the region over which hyperbolicity is lost.
