Existence and profile covariance of the modified scalar scattering matrix and interacting fields

Let $\boldsymbol{\eta}$ be a profile, let $C\in\mathcal{F}$ be a polynomial in fields and their derivatives, and let $\Psi,\Psi'\in\mathcal{D}_2$. A renormalization scheme for time-ordered products is sought such that the adiabatic limits defining $S_{\mathrm{mod}}(\eta)$ and $C_{\mathrm{ret}/\mathrm{adv},\mathrm{mod}}(\eta;x)$ exist in the stated formal distributional spaces. Existence and covariance conjecture. There exists such a renormalization scheme for which

$$(\Psi|S_{\mathrm{mod}}(\eta)\Psi'):=\lim_{\epsilon\searrow0}(\Psi|S_{\mathrm{mod}}(\eta,g_\epsilon)\Psi')$$

exists in $\mathbb{C}\llbracket e\rrbracket$ and defines $S_{\mathrm{mod}}(\eta)\in L(\mathcal{D}_2,\mathcal{D}_2^*)\llbracket e\rrbracket$, while the corresponding limits for every $h\in\mathcal{S}(\mathbb{R}^4)$ exist in $\mathcal{S}'(\mathbb{R}^4)\llbracket e\rrbracket$ and define $C_{\mathrm{ret}/\mathrm{adv},\mathrm{mod}}(\eta;x)\in\mathcal{S}'(\mathbb{R}^4,L(\mathcal{D}_2,\mathcal{D}_2^*))\llbracket e\rrbracket$. For any two profiles $\eta,\eta'$, these objects satisfy

$$S_{\mathrm{mod}}(\eta')=V_{\mathrm{out}}(\eta',\eta)S_{\mathrm{mod}}(\eta)V_{\mathrm{in}}(\eta,\eta'),$$

and

$$C_{\mathrm{ret}/\mathrm{adv},\mathrm{mod}}(\eta';x)=V_{\mathrm{in}/\mathrm{out}}(\eta',\eta)C_{\mathrm{ret}/\mathrm{adv},\mathrm{mod}}(\eta;x)V_{\mathrm{in}/\mathrm{out}}(\eta,\eta').$$

This is presented as the desired construction of the modified scattering matrix and interacting fields in the scalar model; the supplied text gives no evidence that the assertion has been proved or disproved.