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

Let $\eta$ be a profile, let $\mathrm{v}$ be a four-velocity, let $C\in\{F^{\mu\nu},j^\mu\}$ be an observable field, and let $\Psi,\Psi'\in\mathcal{D}_2$. The BRST charge is denoted by $Q_{\mathrm{BRST}}$, and $S_{\mathrm{mod}}(\eta,\mathrm{v})$ and $C_{\mathrm{ret}/\mathrm{adv},\mathrm{mod}}(\eta,\mathrm{v};h)$ are forms on the indicated spaces. Existence and covariance conjecture. There exists a renormalization scheme for time-ordered products such that the adiabatic limits defining the modified QED scattering form and interacting observable fields exist in the stated formal spaces, and, for any two profiles $\eta,\eta'$,

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

Moreover, these forms commute with $Q_{\mathrm{BRST}}$ and therefore induce unique forms on the physical spaces, identified with the physical scattering matrix and physical interacting field. The supplied text gives no evidence that this assertion has been proved or disproved.