  We point out that the stringent lower bounds on the masses of additional
electrically neutral and charged Higgs bosons crucially depend on the flavour
structure of their Yukawa interactions. We show that these bounds can easily be
evaded by the introduction of flavour-changing neutral currents in the Higgs
sector. As an illustration, we study the phenomenology of a two Higgs doublet
model with a Yukawa texture singling out the third family of quarks and
leptons. We combine constraints from low-energy flavour physics measurements,
LHC measurements of the 125 GeV Higgs boson rates, and LHC searches for new
heavy Higgs bosons. We propose novel LHC searches that could be performed in
the coming years to unravel the existence of these new Higgs bosons.
