Time crystals: Can diamagnetic currents drive a charge density wave into rotation?

It has been argued recently that an inhomogeneous system could rotate spontaneously in its ground state - hence a 'time crystal' which is periodic in time. In this note we present a very simple example: a superfluid ring threaded by a magnetic field which develops a charge density wave (CDW). A naive calculation shows that diamagnetic currents cannot drive rotation of the CDW, with a clear picture of the cancellation mechanism.