This isn't true. For example, at the bending magnets in a particle accelerator there is a constant current of electrons moving through the bend but those electrons still radiate. From far away the charge density at the turn is constant, but the current density is not constant. You cannot eliminate current density by squinting - classically accelerating charge radiates.

If we have a classical scale beam of electrons released into a constant magnetic field it radiates as it moves in a circle, even if the instantaneous charge density at any point on the circle is constant. In fact, speaking purely classically, the current density is instantaneously constant at a given point on the circle but it still radiates because in order to maintain itself on a circular path the charge must accelerate, even if the speed stays the same.

Of course as the action of the system approaches hbar (as it must as the energy radiates away) the system begins to have quantized energy levels, as quantum mechanics correctly predicts.