Why can't fermions be affected by effective gravity in non-linear quantum electrodynamics?

Quantum electrodynamics based upon Euler-Heisenberg or Born-Infeld Lagrangians predict photons to move according to an effective metric which is dependent on the background electromagnetic field. In other words, photon trajectories are curved in presence of electromagnetic fields, meaning that an effective gravity is acting upon. If part of fermion masses is allegedly of electromagnetic origin, the question why their trajectories are not affected by this effective gravity naturally comes to mind.