Ah you're right by definition a conductor does have a current provided there's any electrical field inside it (divergent or not). So indeed they can't have char

Ah you're right by definition a conductor does have a current provided there's any electrical field inside it (divergent or not). So indeed they can't have charges inside them in equilibrium. However if you induce an electrical field in a vacuum then no charges at all will flow because a vacuum is the perfect insulator. And even if there is nonzero electric field in and around an individual electron they won't move as long as things cancel out at their exact position. Really the theory of conductors and charge densities seems to break down somewhat once you get to the point of individual electrons near absolute zero in a vacuum. One way to put this to the test would be to charge a conductor to its absolute limit, basically removing all free charges from it and seeing if the rest will crystalize. This might require an impractical amount of energy.