Generalised conformal higher-spin fields in curved backgrounds

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.