Partial masslessness and conformal gravity

We use conformal, but ghostful, Weyl gravity to study its ghost-free, second derivative, partially massless (PM) spin-2 component in the presence of Einstein gravity with positive cosmological constant. Specifically, we consider both gravitational-and self-interactions of PM via the fully nonlinear factorization of conformal gravity's Bach tensor into Einstein times Schouten operators. We find that extending PM beyond linear order suffers from familiar higher spin consistency obstructions: it propagates only in Einstein backgrounds, and the conformal gravity route generates only the usual safe, Noether, cubic order vertices.