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We update the allowed parameter space of the CP-violating 2HDM with Type IIYukawa couplings, that survives the current experimental and theoreticalconstraints on the model. For a representative set of allowed parameter points,we study the production of charged Higgs bosons, both at the LHC at 14 TeV andat a possible future hadronic collider at 30 TeV. Two classes of productionmechanisms are considered, "bosonic" (pp\to H^\pm W^\mp X) and "fermionic" (pp\to H^+ \bar t (b) X). After commenting on our previous H^\pm\to W^\pm H_1investigation, we focus on the tauonic decay mode, H^\pm \to \tau\nu,performing a detailed signal-over-background analysis at the parton level. Theincreased features provided when considering CP violation, i.e., the extensionof the parameter space and the mixing of the would-be CP-odd scalar boson, onlymarginally increase the discovery prospects, which remain very challenging bothwhen increased luminosities and higher energies are considered.

Charged-Higgs production in the Two-Higgs-Doublet Model --- the \tau\nu channel