Residual gauge symmetry of linearized gravity
= Residual gauge symmetry of linearized gravity
An infinitesimal coordinate change generated by $\xi_\mu$ changes
$$
\bar h_{\mu\nu}\mapsto
\bar h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu
+\eta_{\mu\nu}\partial_\rho\xi^\rho.
$$
It preserves Lorenz gauge exactly when $\Box\xi_\mu=0$.