Equality of total inner and outer masses for exhausting legal surfaces

Let $(M^3,g,h)$ satisfy $\mu \ge |J|$, and suppose there is a nested sequence of connected legal surfaces

$$\Sigma_i=\partial D_i\subset M,\qquad 1\le i<\infty,$$

with $\bigcup_iD_i=M$ and $\lim |\Sigma_i|=\infty$. Here legal means outer-minimising and admitting an asymptotically flat extension satisfying $\mu \ge |J|$ outside the surface in which it remains outer-minimising. The legal-surface total-mass conjecture. Then

$$m_{inner}^{total}=m_{outer}^{total}\in\mathbb{R}\cup\{\infty\}.$$

This conjecture addresses when the total mass interval is single-valued beyond the asymptotically flat setting. The supplied text presents it as a belief and gives no resolution evidence.