  The problem of constructing gauge-invariant actions for conformal higher-spin
fields in curved backgrounds is known to be notoriously difficult. In this
paper we present gauge-invariant models for conformal maximal depth fields with
spin $s=5/2$ and $s=3$ in four-dimensional Bach-flat backgrounds. We find that
certain lower-spin fields must be introduced to ensure gauge invariance when
$s>2$, which is analogous to a conjecture made earlier in the literature for
conformal higher-spin fields of minimal depth.
