Does semi-classical gravity obey the equivalence principle?

Question

Question

I was recently wondering about semi-classical gravity :

\(G_{\mu \nu} = \frac{8 \pi G}{c^4} \langle \hat T_{\mu \nu} \rangle_\psi\)

Does this obey the equivalence principle ?

My intuition

The moral of Einsteins elevator thought experiment was (as I understand it): "Given only observables of the lift world one cannot distinguish between gravity and acceleration in free space." If one knows the Hamiltonian one can use Heisenberg equation of motion to define acceleration and it's standard deviation. But if one solves the Einstein tensor and tries find the acceleration one will not find any standard deviation. But this view would only work in quantum mechanics and I'm not sure how to argue it for fields. Does this kind of argument still hold in QFT?