We classify all the possible parity-invariant, non-Abelian couplings involving a partially massless spin-2 field and a pair of massless spin-3/2 fields around anti-de Sitter spacetime AdS4. We re-derive a non-Abelian vertex recently found in [48] by Yu. M. Zinoviev following a different approach, and make explicit the structure of the non-Abelian gauge algebra, very similar to \( \mathcal{N} \) = 2 supergravity around AdS4. We show that this vertex is obstructed at higher order in deformation, and that the graviton plus a vector gauge field are required to eliminate most, if not all, of the obstructions. The resulting gauge algebra produces all the structures of the super algebra su(2, 2|1), suggesting that the only consistent theory that couples a partially massless spin-2 field with massless gravitini is \( \mathcal{N} \) = 1 conformal supergravity expanded around AdS4.
