= Solution
The local <gauge symmetry> contains the constant-phase subgroup. For constant $\alpha$, $\delta A_\mu=0$, and <Noether's theorem> applied to the matter fields gives
$$
\frac{j_{\rm N}^\mu}{\alpha}
=(\bar D^\mu\phi^*)(-iq\phi)+(D^\mu\phi)(iq\phi^*)=J^\mu.
$$
Thus $\partial_\mu J^\mu=0$ on the matter field equations. More explicitly, allowing $\alpha(x)$ in the matter variation while holding $A$ fixed gives $\delta\mathcal L=J^\mu\partial_\mu\alpha$. On shell it also equals $\partial_\mu(\alpha J^\mu)$, so equality for arbitrary $\alpha$ implies conservation. In the full gauge transformation, varying $A$ cancels the $J^\mu\partial_\mu\alpha$ term; this is the related Noether second-theorem identity, not a new independent charge for every gauge function.
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