Limiting Riemannian Penrose inequality conjecture for constrained maximizers

Let $(M,g)$ be an asymptotically flat $n$-manifold, with $3 \leq n \leq 7$, nonnegative scalar curvature, and nonempty, smooth, compact boundary $\Sigma$. Fix $A>0$ such that $\alpha(A)<A$, and let $\overline g_C$ be a maximizer for $\alpha_C(A)$.

Limiting Riemannian Penrose inequality conjecture. Then

$$\lim_{C \to \infty} m_{ADM}(\overline g_C) \geq \lim_{C \to \infty} \frac{1}{2}\left(\frac{\min(\Sigma,\overline g_C)}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}}.$$

In other words, the Riemannian Penrose inequality should hold for $(M,\overline g_C)$ in the limit $C \to \infty$. The conjecture is motivated by the known Riemannian Penrose inequality in dimensions $3$ through $7$ and by the expected near-minimal behavior of the outermost minimal area enclosure for large $C$; the limiting assertion remains open in the supplied source.