“…In order to illustrate such a phenomenology, we have defined several BPs, each embedding CP-violation, over the I(2+1)HDM parameter space, with varying mass splittings and coupling strengths in the inert sector, all compliant with available experimental data, from relic density, (in)direct DM searches and colliders. For three such BPs, we have quantified CP-violation effects entering three asymmetries which can all be defined in the qq → Z * → ZZ channel and measured at both the LHC (f = q) by the end of its lifetime (i.e., after the High Luminosity LHC [58,59] runs) and at future lepton colliders (f = e) such as the FCC-ee, ILC, CLiC or CEPC running at current design luminosities. Finally, we have illustrated that the hadronic cross sections are typically larger than the leptonic ones, so that it is quite possible that a first evidence of a CPviolating I(2+1)HDM will occur at the LHC rather than at the FCC-ee, ILC, CLiC or CEPC.…”