The simplest supergravity theory (N=1, D=4) has the gravity part expressed in the so-called vielbein-spin connection formalism. Without vielbeins, you cannot properly put spinorial fields (Weyl, Dirac, Rarita-Schwinger) in curved spacetimes. Therefore, the Lagrangian density and the integration measure look different, the field equations of the combined HE-RS look different.

Boulanger and Esole (https://arxiv.org/abs/gr-qc/0110072) have proven the uniqueness of the SUGRA N=1, D=4 starting by BRST-deforming spin-2 fields and spin-3/2 fields in flat spacetime. The linearized vielbein is essentially the spin-2 Pauli-Fierz field (i.e. the 1st order perturbation of the metric).

The simplest supergravity theory (N=1, D=4) has the gravity part expressed in the so-called vielbein-spin connection formalism. Without vielbeins, you cannot properly put spinorial fields (Weyl, Dirac, Rarita-Schwinger) in curved spacetimes [chapter #13 of Wald's book]. Therefore, the Lagrangian density and the integration measure look different, the field equations of the combined HE-RS look different. The Wikipedia article on this uses the superspace formalism.

At an elementary level, Boulanger and Esole (https://arxiv.org/abs/gr-qc/0110072) have proven the uniqueness of the SUGRA N=1, D=4 in curved bosonic spacetime starting by BRST-deforming spin-2 fields and spin-3/2 fields in flat spacetime. The linearized vielbein is essentially the spin-2 Pauli-Fierz field (i.e. the 1st order perturbation of the metric).