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.
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