I posted this question on Stack Exchange Physics (https://physics.stackexchange.com/questions/679845/do-these-greens-functions-satisfy-the-lorenz-gauge-condition), but repost it here since I didn't get any answers, and it is quite mathematical.
I have read a little in "The wave equation on a curved space-time" by Friedlander. The book gives a construction of the retarded and advanced green's functions of a hyperbolic wave operator on a lorentzian manifold with main focus on the 4-dimensional case.
I want to find the advanced/retarded green's function to solve Maxwell's equation in curved spacetime $(M,g)$ which is a 4-dimesnional Lorentzian manifold of signature $(+---)$. Maxwell's equation reads $$\Box A_\mu-\partial_\mu\left(\nabla_\nu A^\nu\right)-R_{\mu\nu}A^\nu=J_\mu.$$ To make this on the hyperbolic form that Friedland's book requires, one can impose the Lorenz gauge condition $$\nabla_\nu A^\nu=0.$$ If one fixes a $p\in M$ and restricts oneself to a causal domain $\Omega_0\subseteq M$ (p. 146) around $p$, then a solution to $$\Box A_\mu-R_{\mu\nu}A^\nu=J_\mu$$ can be found by the two green's functions $$G^\pm_{\mu\nu}(p,q)=\frac{1}{2\pi}\kappa(p,q)\tau_{\mu\nu}(p,q)\delta^{\pm}\left(\Gamma(p,q)\right) + \frac{1}{2\pi}V^\pm_{\mu\nu}(p,q)$$ where $$ \begin{align} \Gamma(p,q)&:= \langle X,X\rangle \text{ where }\langle,\rangle \text{ is the metric tensor and } p=\exp_q(X) \\ \kappa(p,q)&:=\lvert\det g\rvert^{-1/4}, \text{ where the metric } g \text{ is written in normal coordinates around }p\text{ and evaluated at } q \\ \tau_{\mu\nu}(p,q)& \text{ is parallel transport along the unique geodesic between } p \text{ and } q \\ \delta^{\pm}\left(\Gamma(p,q)\right)&:=\lim_{\epsilon\downarrow 0} \chi_{\mathcal{J}^\pm(p)}(q)\cdot\delta\left(\Gamma(p,q)-\epsilon\right) \text{ and } \delta \text{ is the Dirac delta distribution on } \mathbb{R} \\ \mathcal{J}^\pm(p)& \text{ are the future and past points with causal geodesic to } p \text{ and } \chi_{\mathcal{J}^\pm(p)} \text{ its indicator function} \\ V^\pm_{\mu\nu}(p,q)& \text{ is a smooth function with support in } \left\{(p,q)\in\Omega_0\times\Omega_0\mid q\in\mathcal{J}^\pm(p)\right\} \end{align} $$ The construction of $V^\pm$ is described in Friedlander's book.
The question is: if $\nabla_\mu J^\mu=0$, how does one show whether $\nabla_\mu A^\mu=0$ where $$\langle A_\mu(p), \phi^\mu(p) \rangle:=\langle \langle G^+_{\mu\nu}(p,q), \phi^\nu(q)\rangle, J^\mu(p)\rangle \qquad\text{ and }\phi\text{ is a test function on }\Omega_0~?$$
From the reciprocity thm. in the book and the fact that the wave operator is self-adjoint, $G^+_{\mu\nu}(p,q)=G^-_{\nu\mu}(q,p)$ so $A_\mu(p)=\lim_{n\rightarrow\infty}\langle G^-_{\mu\nu}(p,q), J_n^\nu(q)\rangle$ where $\left\{J_n\right\}_n$ is any sequence of test functions converging to $J$.