= Solution
For a <torsion-free connection>, the connection terms in the antisymmetrized <covariant derivative> cancel: $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. Thus the <electromagnetic field tensor> is the <exterior derivative> of the <four-potential>. It is a <tensor>, is invariant under $A_\mu\mapsto A_\mu+\partial_\mu\chi$, and obeys the homogeneous <Maxwell equations> $\nabla_{[\alpha}F_{\beta\gamma]}=0$. The covariant divergence equation is the generally covariant form of the local source equation; use its minus sign as specified in the paper.
Expand that equation before imposing the <Lorenz gauge condition>:
$$
\nabla^\mu F_{\mu\nu}=\nabla^\mu\nabla_\mu A_\nu-\nabla^\mu\nabla_\nu A_\mu.
$$
The <curvature commutator on a covariant tensor> gives
$$
[\nabla_\rho,\nabla_\nu]A_\mu=-R^\alpha{}_{\mu\rho\nu}A_\alpha,
\qquad
\nabla^\mu\nabla_\nu A_\mu=\nabla_\nu(\nabla^\mu A_\mu)+R^\alpha{}_\nu A_\alpha.
$$
Here the contraction $g^{\mu\rho}R^\alpha{}_{\mu\rho\nu}=-R^\alpha{}_\nu$ accounts for the second sign reversal. Hence in <Lorenz gauge>
$$
\boxed{\nabla^\mu\nabla_\mu A_\nu-R^\alpha{}_\nu A_\alpha=-\mu_0J_\nu.}
$$
This <curved-spacetime electromagnetic potential wave equation> contains a <Ricci tensor> term even in a locally inertial coordinate system: a connection can vanish at one point while its derivatives and <curvature> remain nonzero.
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