Remark: The solution-space mismatch is substantive

The wave equation $\square h=0$ is not the vacuum Einstein equation $\mathcal{L}h=0$ on unrestricted tensor fields. For example, let

Then $\square h=0$, whereas direct substitution gives

Conversely, every compact pure gauge $K\xi$ belongs to ${\mathcal{S}}_{\mathrm{E}}$, but a nonzero compact pure gauge cannot solve the homogeneous wave equation, by uniqueness for zero past data. Thus an on-shell equality on ${\mathcal{S}}_{\square }$ does not automatically give an equality on ${\mathcal{S}}_{\mathrm{E}}$.