Conjecture on smooth parametrization by initial, boundary, and corner data

Let ${\mathcal E}$ be the space of solutions modulo the relevant gauge equivalence, let ${\mathcal B}={\mathcal B}_C$ denote the conformal-mean curvature boundary data space, let ${\mathcal I}_0$ denote the initial data space, and let ${\mathcal A}=C^{\infty}(\Sigma)$ be the space of smooth functions on the corner $\Sigma$. For a solution $g$, write $(g_S,K_S)$ for its initial data, $[g_{\mathcal C}]$ for the conformal class of the induced boundary metric, $H_{\mathcal C}$ for the boundary mean curvature, and $\alpha_g$ for its corner data. Conjecture on smooth parametrization by initial, boundary, and corner data. The space ${\mathcal E}$ is a smooth (Fréchet) manifold, and the smooth map

$$\Phi:{\mathcal E}\longrightarrow {\mathcal I}_0\times_c{\mathcal B}\times{\mathcal A},$$ $$\Phi(g)=((g_S,K_S),[g_{\mathcal C}],H_{\mathcal C},\alpha_g)$$

is a diffeomorphism. Thus, after adding corner data, the initial-boundary value constraint problem is locally-in-time well-posed. The conjecture is motivated by non-isometric families with identical initial and conformal-mean curvature boundary data but varying corner angle; its resolution would provide an effective local parametrization of solutions by the augmented data.