Can Maxwell’s equations be obtained from the continuity equation?

The evaluation at the retarded time must be interpreted unambiguously. The quantity $[∇′∙J]$ means that we first calculate the divergence of the current density at the present time $∇′∙J(x′,t)$ and in the resulting expression $f1(x′,t)$ we replace $t$ by the retarded time $t′=t−R∕c$⁠, that is, $f(x′,t′)$⁠. Analogously, the quantity $[∂ρ∕∂t]$ means that we first calculate the time derivative of the charge density at the present time $∂ρ(x′,t)∕∂t$ and in the resulting expression $f2(x′,t)$ we replace $t$ by the retarded time $t′=t−R∕c$⁠, that is, $f2(x′,t′)$⁠. We also note that $[∂ρ∕∂t]=[∂ρ∕∂t′]$⁠, which follows from the result $∂ρ(x′,t′)∕∂t=(∂ρ(x′,t′)∕∂t′)(∂t′∕∂t)=∂ρ(x′,t′)∕∂t′$⁠.

This nice interpretation of Eq. (5) was suggested by an anonymous referee.

We could also consider sources that vanish sufficiently at infinity, that is, sources of order $O∣x∣−2−δ$⁠, where $δ>0$ as $∣x∣$ goes to infinity, $x$ being the field point.