  We continue the investigation of the structure of the action for a tower of
conformal higher spin fields in non-trivial 4d background metric recently
discussed in arXiv:1609.09381. The action is defined as an induced one from
path integral of a conformal scalar field in curved background coupled to
higher spin fields. We analyse in detail the dependence of the quadratic part
of the induced action on the spin 1 and spin 3 fields, determining the presence
of a curvature-dependent mixed spin 1 - spin 3 term. One consequence is that
the pure spin 3 kinetic term cannot be gauge-invariant on its own beyond the
leading term in small curvature expansion. We also compute the non-zero
contribution of the 1-3 mixing term to the conformal anomaly c-coefficient. One
is thus to determine all such mixing terms before addressing the question of
possible vanishing of the total c-coefficient in the conformal higher spin
theory.
