  Motivated by constant-G theory, we introduce a one-parameter family of
scalar-tensor theories as an extension of constant-G theory in which the
conformal symmetry is a cosmological attractor. Since the model has the
coupling function of negative curvature, we expect spontaneous scalarization
occurs and the parameter is constrained by pulsar-timing measurements. Modeling
neutron stars with realistic equation of states, we study the structure of
neutron stars and calculate the effective scalar coupling with the neutron star
in these theories. We find that within the parameter region where the
observational constraints are satisfied, the effective scalar coupling almost
coincides with that derived using the quadratic model with the same curvature.
This indicates that the constraints obtained by the quadratic model will be
used to limit the curvature of the coupling function universally in the future.
