  We investigate asymptotic safety of a toy model of a singlet-scalar extension
of the Higgs sector including two real scalar fields under the impact of
quantum-gravity fluctuations. Employing functional renormalization group
techniques, we search for fixed points of the system which provide a tentative
ultraviolet completion of the system. We find that in a particular regime of
the gravitational parameter space the canonically marginal and relevant
couplings in the scalar sector -- including the mass parameters -- become
irrelevant at the ultraviolet fixed point. The infrared potential for the two
scalars that can be reached from that fixed point is fully predicted and
features no free parameters. In the remainder of the gravitational parameter
space, the values of the quartic couplings in our model are predicted. In light
of these results, we discuss whether the singlet-scalar could be a dark-matter
candidate. Furthermore, we highlight how "classical scale invariance" in the
sense of a flat potential of the scalar sector at the Planck scale could arise
as a consequence of asymptotic safety.
