Fermion propagator from scalar propagator in curved space

Question

In Minkowski spacetime, the Dirac equation can be found from the scalar propagator by the formula

\(S_F (x - x') = (i \gamma^\mu \partial_\mu + m) G_F (x - x')\)

Does a similar formula hold in curved spacetime? That is, can we just replace the partial derivative by the appropriate covariant derivative and arrive at the spinor propagator that way? I'm a little suspicious because of the extra curvature term that appears when squaring the Dirac operator:

\((\gamma^\mu \nabla_\mu)^2 = \nabla^\mu \nabla_\mu - R/4\)

If the above formula is wrong, is there a correct way to get the spinor propagator from the scalar propagator in an arbitrary spacetime?

Comments

you will find the standard vierbein formulation in Fermions in gravity with local spin-base invariance , page 2 the 20 first lines. "once a suitable vierbein is introduced, the spin connection required to define fermionic dynamics is derived from the vierbein postulate" etc ...