Conjecture on large strictly stable STCMC surfaces in asymptotically Schwarzschildean lightcones

Let $\mathcal{N}$ be an asymptotically Schwarzschildean lightcone. Let $\Sigma$ be a strictly stable STCMC surface with area radius $\rho$, and let $B_\sigma(B_1,B_2,B_3)$ denote the a-priori class used in the paper. Large-surface gauge conjecture. Any strictly stable STCMC surface $\Sigma$ lies in the a-priori class $B_\sigma(B_1,B_2,B_3)$ for suitable values of $B_1,B_2,B_3$ if its area radius $\rho$ is sufficiently large. This would justify the chosen gauge and, if true, allow the main foliation result to be stated in a gauge-independent geometric form. The conjecture is motivated by the corresponding result in the Schwarzschild lightcone, while the source notes that analogous Riemannian counterexamples make such a statement unavailable under weaker asymptotics without additional conditions.