Unique constant-curvature simultaneity distribution for conformally flat polar ends

Let $(M,\mathcal{C})$ be a Riemann-Lorentz conformal space with polar end $D^{\infty}$. Suppose that $D^{\infty}$ is conformal-flat. A simultaneity distribution is a distribution whose integral hypersurfaces represent simultaneity hypersurfaces. The polar-end constant-curvature conjecture. There exists, locally, a simultaneity distribution $\mathcal{D}$ of constant curvature, and $\mathcal{D}$ is univocally determined. Equivalently, the equation $(\tau=0)$ of $D^{\infty}$ is univocally determined by the constant-curvature condition up to diffeomorphism. The conjecture expresses the claim that a conformally flat polar end determines a unique cosmologically relevant constant-curvature simultaneity structure, locally and up to diffeomorphism.