Does the electric force on a charged particle in a uniform electric field increase?

If I have a proton in a uniform field between two parallel oppositely charged plates and the proton accelerates, the electric force acting on it stays constant seeing it is a uniform field and as a result the acceleration of the particle is also constant.

But does the electric force on the proton increase as it nears the negative plate? I would think yes seeing the distance is getting smaller (cf. Coulomb's law) so the electric force (which originates from Coulomb interactions between the charged particle and the plates) must be getting bigger. But this conflicts with the fact that the force is constant. Any help?

                             as  E=F/Q