Corollary: The one-slot rewriting is the case n=1

$\ker {\mathsf{E}}_{\mu }\cap {\mathcal{E}}^{\prime }(M)=(\square +{\mu }^2){\mathcal{E}}^{\prime }(M)$, and every $w\in {\mathcal{E}}^{\prime }(M)$ agrees on solutions with $(\square +{\mu }^2)[\chi {\mathsf{E}}_{\mu }w]$, compactly supported, with $\mathrm{WF}\subset \mathrm{WF}({\mathsf{E}}_{\mu }w)$. For a boundary trace $w=a\delta (t)$, $a\in {C_c}^{\infty }({\mathbb{R}}^3)$, ${\mathsf{E}}_{\mu }w$ is the smooth solution with Cauchy data $(0,a)$, so the representative is smooth and flat at $\Sigma$ after the cutoff. This is XI-F's remark. It does not address two slots, where ${\mathsf{W}}_u$ is a bisolution in the two metric variables and its Cauchy data on $\Sigma \times \Sigma$ are not smooth.