= Solution
Treat each $x^\alpha$ as a <scalar field>. Its second <covariant derivative> is $\nabla_\beta\nabla_\gamma x^\alpha=-\Gamma^\alpha{}_{\gamma\beta}$, so $\Box x^\alpha=-H^\alpha$, where $H^\alpha=g^{\beta\gamma}\Gamma^\alpha{}_{\beta\gamma}$.
On the other hand, <metric compatibility> gives
$$
0=\nabla_\nu g^{\mu\nu}=\partial_\nu g^{\mu\nu}+\Gamma^\mu{}_{\lambda\nu}g^{\lambda\nu}+\Gamma^\nu{}_{\lambda\nu}g^{\mu\lambda}.
$$
Using the <Christoffel trace identity> in the last term yields
$$
\partial_\nu(\sqrt{-g}\,g^{\mu\nu})=-\sqrt{-g}\,H^\mu.
$$
Since the volume density never vanishes in a valid chart, \b[the harmonic coordinate conditions are equivalent]:
$$
\boxed{\Box x^\alpha=0\iff g^{\beta\gamma}\Gamma^\alpha{}_{\beta\gamma}=0
\iff\partial_\nu(\sqrt{-g}\,g^{\mu\nu})=0.}
$$
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