  The complex two-Higgs doublet model (C2HDM) is one of the simplest extensions
of the Standard Model with a source of CP-violation in the scalar sector. It
has a $\mathbb{Z}_2$ symmetry, softly broken by a complex coefficient. There
are four ways to implement this symmetry in the fermion sector, leading to
models known as Type-I, Type-II, Lepton-Specific and Flipped. In the latter
three models, there is a priori the surprising possibility that the 125 GeV
Higgs boson couples mostly as a scalar to top quarks, while it couples mostly
as a pseudoscalar to bottom quarks. This ``maximal'' scenario was still
possible with the data available in 2017. Since then, there have been more data
on the 125 GeV Higgs boson, direct searches for CP-violation in angular
correlations of $\tau$-leptons produced in Higgs boson decays, new results on
the electron electric dipole moment, new constraints from LHC searches for
additional Higgs bosons and new results on $b\to s \gamma$ transitions.
Highlighting the crucial importance of the physics results of LHC's Run 2, we
combine all these experiments and show that the ``maximal'' scenario is now
excluded in all models. Still, one can have a pseudoscalar component in $h
\tau\bar{\tau}$ couplings in the Lepton-Specific case as large as $87\%$ of the
scalar component for all mass orderings of the neutral scalar bosons.
