  In order to address the observation of the neutrino oscillations and the
metastability of the Standard Model, we extend the fermion sector with two
right-handed (i.e. sterile) neutrinos, and the scalar sector of the SM with a
real scalar, the Hill field. The latter takes the role of a Majoron and
generates the Majorana masses for the neutrino sector, such that the particle
spectrum features two CP-even scalars $h_1$ and $h_2$, and also two heavy, mass
degenerate neutrinos. When the $h_1$ is identified with the scalar resonance at
~125 GeV and the condition is imposed that the $h_1$ self coupling and its
running vanish at the Planck scale, the scalar mixing and the vacuum
expectation value of the Hill field are fixed by the $h_2$ mass. The $h_2$ can
be searched for at the LHC, and it has prospects of being discovered for the
target integrated luminosities of the HL-LHC and the Future Circular hadron
Collider (FCC-hh) when its mass is on the weak scale. The knowledge of the
$h_2$ mass and its decay properties can yield an insight into its coupling to
the heavy neutrinos, and thus also on the heavy neutrino mass scale. This
yields an interesting connection between potentially detectable heavy scalars
in high-energy proton collisions and the mass scale of the heavy neutrinos that
is testable at the LHC and at future colliders.
