This Aharonov-Bohm effect has been observed. It is a purely quantum mechanical effect. Classically the motion of the particle is determined by Newton's laws and the Lorentz force is zero in regions where B is zero. In quantum mechanics the phase of the wave function depends on the vector potential A . However the observable effects depend on phase differences, which in turn depend on the flux of the magnetic field B .
Observable effects have to be gauge invariant.
Y. Aharonov and D. Bohm, Phys.