Does a charged particle fall as fast as a neutral particle?

https://en.wikipedia.org/wiki/Paradox_of_radiation_of_charged_particles_in_a_gravitational_field

Let us try to solve the paradox. Fritz Rohrlich (1965) claims that a charged particle and a neutral particle fall as fast in a gravitational field. Is that true?

Let us model the charged particle and its electric field with an elastic extended object. The object extends far away, outside the gravitational field. Now it is obvious that the charged particle will not fall as fast as a neutral particle. The parts of the electric field outside the gravitational field will slow down the fall.

Does this break the equivalence principle? No. The system in a falling laboratory is not isolated from its environment. It interacts with the electric field which extends to the infinity.

The interaction with the outer parts of the electric field is small, but it is not small compared to the interaction when we oscillate the charge to create long electromagnetic waves. Thus, we must consider the interaction in our analysis - we cannot ignore it.