Gauge transformations with varying phase give us conservation of the charge density. Hence charged particles cannot move?

I stumbled upon the following paragraph in Quark confinement and Topology of gauge theories by Polyakov

"Gauge invariance with constant phase $$\Psi \to e^{i \alpha}$$ lead to conservation of the total charge. Gauge transformations with a varying phase $$\Psi \to e^{i \alpha(x)}$$ will give us conservation of the charge density. But this in term means that the charged particle cannot move. The only thing which saves the electron from this fatal immobility is the degeneracy of the vacuum in QED, that is, its non invariant under gauge transformations."

Are these statements correct? For example, I've never heard before that charge density is conserved due to local gauge invariance. Or that the QED vacuum isn't invariant under gauge transformations.

(The paper has almost 1500 citations, so I suppose his statements are correct. But I have never seen them anywhere else or any concrete calculations which backs them up.)

As Qmechanic elucidates in detail in this excellent answer, the "second Noether current" vanishes off-shell and its charge is identically zero under reasonable assumptions, and for electrodynamics it is the trivial statement that $$\partial_\mu \partial_\nu F^{\mu\nu} = 0$$.