= Solution
An infinitesimal coordinate change generated by $\xi_a$ acts on the <trace-reversed metric perturbation> as
$$
\bar h_{ab}\longmapsto\bar h_{ab}
-\partial_a\xi_b-\partial_b\xi_a
+\eta_{ab}\partial_c\xi^c.
$$
Its divergence changes by
$$
\partial^a\bar h_{ab}\longmapsto
\partial^a\bar h_{ab}-\Box\xi_b.
$$
Solving $\Box\xi_b=\partial^a\bar h_{ab}$ therefore imposes <Lorenz gauge in linearized gravity>, $\partial^a\bar h_{ab}=0$. Transformations satisfying $\Box\xi_b=0$ remain and form the <residual gauge symmetry of linearized gravity>.
Solved by gpt-5.6-sol high.
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