Hi smallsteps,
The reason it cannot be explained by a classical flux outside of the magnet is held in the results - i.e. the effect that upon the diffraction pattern of the electrons on the screen behind the two slits within the Aharonov-Bohm effect.
For ideal purposes, the B-field of the solenoid is confined purely to the solenoid itself, as in the case of an infinitely long solenoid. In reality of course, this is not possible, and so optimum materials must be chosen in order to approximate this behaviour to the greatest extent possible.
In the Schrodinger equation, if you substitute for the wavefunction associated with each electron a further phase term (based upon the requirement that our Schrodinger equation should not be dependent upon the phase of the wavefunction as the probabilty distribution function that results from it is the magnitude of the wavefunction, not phase dependent), your resultant Schrodinger equation has further components that may be equated to magnetic vector potential A and scalar potential V. To see this is action, see
http://www.vttoth.com/gauge.htm for an example.
If you treat the B field outside of the solenoid as zero as in the case of the infinitely long solenoid, then use of QM (outlined in the original Aharonov-Bohm paper) shows that the electrons are behaving as if their momentum is perturbed by an extra component involving a magnetic vector potential A. This is the ideal situation, and it correctly predicts experimental results in the real situation.
Classically, we would only expect the electrons to be perturbed by a curl component of the B field = grad x A, but since the B field is zero, such A is not the result of the B-field within the solenoid. Indeed, classically, it does not exist outside of the solenoid.
Note also that in the Lorentz force, we expect the electrons classically to be perturbed by a v x B component. But B=0, and even if scientists
had made the mistake of not measuring a weak B-field outside of their experimental solenoid, the A component does not have the same direction (or magnitude) of the v x B component in the Lorentz force, since B = grad x A.
My further argument is non-scientific. I simply don't think using common sense that the thousands of scientists that have understood and written papers using quantum AB effect as a theoretical basis have simply got their classical EM wrong.
Incidentyl, note that the classical concept of the electron being subject to a Lorentz force, indeed the classical concept of the electron particle without associated wavefunction determining the statistical probability of its behaviour, does not explain why even a diffraction pattern appears if you fire at the two slits in the first place!