Consider a theory of a free massless complex scalar $\phi$ which undergoes global $U(1)$ transformations. The conserved current associated to this symmetry is the usual scalar current
$$ J^\mu = i\left(\phi^\dagger \partial^\mu \phi - \partial^\mu\phi^\dagger \phi\right) \tag{1} $$
which is divergence-less on-shell: $$\partial_\mu J^\mu = \phi^\dagger \square \phi -h.c = 0 \tag{2}$$ since $$\square\phi=0.\tag{3}$$
When we gauge the $U(1)$, we expect that gauge symmetriy should not spoil global current conservation. The Lagrangian is now
$$ -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - (D_\mu\phi)^\dagger(D_\mu\phi) = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}- (\partial_\mu\phi)^\dagger(\partial_\mu\phi) - A_\mu J^\mu - A_\mu^2\phi^\dagger\phi \tag{4} $$ where $$D_\mu\phi = \partial_\mu \phi -i A_\mu \phi. \tag{5}$$
The equation of motions for the photon are
$$ \partial_\mu F^{\mu\nu} = J^\mu + 2A_\mu\phi^\dagger\phi \tag{6} $$
which seems to imply that the global current is not even conserved, that is
$$ \partial_\mu J^\mu = -2\partial_\mu\left[A^\mu \phi^\dagger \phi\right]. \tag{7} $$
Is this result wrong? It is against the expectations. The theory is still invariant under global $U(1)$ transformations, so the global current should be conserved.
