  In Lagrangian gauge systems, the vector space of global reducibility
parameters forms a module under the Lie algebra of symmetries of the action.
Since the classification of global reducibility parameters is generically
easier than the classification of symmetries of the action, this fact can be
used to constrain the latter when knowing the former. We apply this strategy
and its generalization for the non-Lagrangian setting to the problem of
conformal symmetry of various free higher spin gauge fields. This scheme allows
one to show that, in terms of potentials, massless higher spin gauge fields in
Minkowski space and partially-massless fields in (A)dS space are not conformal
for spin strictly greater than one, while in terms of curvatures, maximal-depth
partially-massless fields in four dimensions are also not conformal, unlike the
closely related, but less constrained, maximal-depth Fradkin--Tseytlin fields.
