It is known that pure electrodynamics in curved space-time is invariant under Weyl transformations

$$

g_{\mu\nu} \to \Omega(x)g_{\mu\nu}, \quad F_{\mu \nu} \to \Omega^{-1}(x)F_{\mu\nu} \qquad (1)

$$

Now let's assume that there is external current $J_{\mu}$. Now an action becomes

$$

S=\int d^{4}x\sqrt{-g}\left( -\frac{1}{4}F_{\mu\nu}F_{\alpha\beta}g^{\mu\nu}g^{\alpha\beta} - J_{\mu}A_{\nu}g^{\mu\nu}\right) \qquad (2)

$$

Is it true that it isn't Weyl invariant? I.e., is it possible to construct the explicit form of the "conformal" transformation of $J_{\mu}$, which preserves Weyl invariance in some generalized sense?

The question appears because of Maxwell equation in FRLW space-time in presence of

$$

J_{\mu} = (0, \sigma \mathbf E) \qquad (3)

$$

This equation takes the form

$$

B{''} + \sigma \mathbf {B}{'} + k^{2}\mathbf B = 0, \qquad (4)

$$

which is completely the same as corresponding equation in flat space-time, which seems to mean that conformal invariance is preserved even in presence of current $J_{\mu}$ $(3)$. This current isn't external, since it is expressed in terms of $\mathbf{E}$, but I can't get an appropriate transformation for $\mathbf A$, so that

$$

\int d^{4}x\sqrt{-g}J_{\mu}A_{\nu}g^{\mu\nu} \to \int d^{4}x\sigma\sqrt{-g}a^{-2}(\mathbf E \cdot \mathbf A)

$$

becomes invariant.