  We complete the proof of the conjecture in arXiv:2111.06903 regarding the
form of gravitational building blocks in higher derivative supergravity and
examine the related gluing rules. The derivation, based on certain assumptions,
combines the BVAB equivariant localization formula (on non-compact spaces) and
the direct evaluation of the action on the Omega background in AdS$_4$,
arXiv:2212.10590. Utilizing the superconformal gravity formalism and avoiding
equations of motion ensures our results are valid at any perturbative order in
the Newton constant. As an application, we illustrate the holographic
predictions for supersymmetric M2-brane partition functions. We conclude with a
to-do list of related calculations in effective supergravity and string theory,
aiming for full quantum control of BPS backgrounds.
