To the question "What is the electric field outside a cylindrical solenoid when inside is turned on a magnetic field" the answer is that outside exists a electric field. Does that mean that the fringes shift in the double slit experiment with electrons could be explained with electromagnetic fields and it is not necessary (but of course possible) to explain it with quantum mechanics?
This is what the author of http://arxiv.org/abs/1407.4826 seems to imply. I have no idea if this is correct or not, sorry.
The issue here isn't quantum mechanics but whether a description using only local fields is possible. Quantum mechanics is essential as you are considering an interference phenomenon. The alternative would be to consder a charged classical field that corresponds to the electrons in a superposition of the two paths, but then you are simply considering the trivial limit of large numbers of particles.
Lev Vaidman has argued here that the non-local effect in the AB-effect is an artifact of describing the solenoid classically. In reality, everything is quantum mechanical including the solenoid. It is then the electric field of the electron that exerts a force on the charges in the solenoid, the momentum transfer leads to the wavefunction of the solenoid to pick up a phase which is different for the two paths and that difference is precisely given by the AB formula involving the flux.