  It has been proposed in \cite{Park:2014tia} that 4D Einstein gravity becomes
effectively reduced to 3D after solving the Lagrangian analogues of the
Hamiltonian and momentum constraints of the Hamiltonian quantization. The
analysis in \cite{Park:2014tia} was carried out at the classical/operator
level. We review the proposal and make a transition to the path integral
account. We then set the stage for explicitly carrying out the two-loop
renormalization procedure of the resulting 3D action. We also address a
potentially subtle issue in the gravity context concerning whether
renormalizability does not depend on the background around which the original
action is expanded.
