Gauge-independent uniqueness conjecture for STCMC foliations

Let $\mathcal{N}$ be an asymptotically Schwarzschildean lightcone. An asymptotic foliation by strictly stable STCMC surfaces is a foliation of the asymptotic region of $\mathcal{N}$ whose leaves are strictly stable spacetime constant-mean-curvature surfaces. Gauge-independent foliation conjecture. There exists a unique asymptotic foliation of $\mathcal{N}$ by strictly stable STCMC surfaces. The claim would provide a gauge-independent, purely geometric formulation of the main result, conditional on the preceding large-surface gauge conjecture. It is refuted as stated by explicit counterexamples of Brendle--Eichmair in the analogous Riemannian setting; the source indicates that only weaker asymptotics together with a suitable energy condition may support an analogous statement.