  Certain scalar-tensor theories of gravity that generalize
Jordan-Fierz-Brans-Dicke theory are known to predict non-trivial phenomenology
for neutron stars. In these theories, first proposed by Damour and
Esposito-Far\`ese, the scalar field has a standard kinetic term, and couples
conformally to the matter fields. The weak equivalence principle is therefore
satisfied, but scalar effects may arise in strong-field regimes, e.g. allowing
for violations of the strong equivalence principle in neutron stars
("spontaneous scalarization") or in sufficiently tight binary neutron-star
systems ("dynamical/induced scalarization"). The original scalar-tensor theory
proposed by Damour and Esposito-Far\`ese is in tension with solar-system
constraints (for couplings that lead to scalarization), if one accounts for
cosmological evolution of the scalar field and no mass term is included in the
action. We here extend the conformal coupling of that theory, in order to
ascertain if, in this way, solar-system tests can be passed, while retaining a
non-trivial phenomenology for neutron stars. We find that even with this
generalized conformal coupling, it is impossible to construct a theory that
passes both Big-Bang nucleosynthesis and solar-system constraints, while
simultaneously allowing for scalarization in isolated/binary neutron stars.
