  The type-II see-saw mechanism based on the annexation of the Standard Model
by weak gauge triplet scalar field proffers a natural explanation for the very
minuteness of neutrino masses. Noting that the phenomenology for the
non-degenerate triplet Higgs spectrum is substantially contrasting than that
for the degenerate one, we perform a comprehensive study for an extensive model
parameter space parametrised by the triplet scalar vacuum expectation value
(VEV), the mass-splitting between the triplet-like doubly and singly charged
scalars and the mass of the doubly charged scalar. Considering all Drell-Yan
production mechanisms for the triplet-like scalars and taking into account the
all-encompassing complexity of their decays, we derive the most stringent 95%
CL lower limits on the mass of the doubly charged scalar for a vast model
parameter space by implementing already existing direct collider searches by
CMS and ATLAS. These estimated limits are beyond those from the existing LHC
searches by approximately 50-230 GeV. However, we also find that a specific
region of the parameter space is not constrained by the LHC searches. Then, we
forecast future limits by extending an ATLAS search at high-luminosity, and we
propose a search strategy that yields improved limits for a part of the
parameter space.
