  While the triplet-like Higgses up to a few hundred GeV masses are already
excluded for a vast region of the model parameter space from the LHC searches,
strikingly, there is a region of this parameter space that is beyond the reach
of the existing LHC searches, and doubly/singly-charged and neutral Higgses as
light as 200 GeV or even lighter are still allowed by the LHC data. We study
several search strategies targeting different parts of this LHC elusive
parameter space at two configurations of $e^-e^+$ colliders -- 500 GeV and 1
TeV centre of mass energies. We find that a vast region of this parameter space
could be probed with 5$\sigma$ discovery with the early $e^-e^+$ colliders'
data.
