Mixed Riemannian Penrose–ZAS inequality

Let $(M,g)$ be an asymptotically flat three-manifold with nonnegative scalar curvature and compact smooth boundary $\partial M=S\cup\Sigma$, where $S$ is an area outer-minimizing minimal surface and $\Sigma$ consists of zero area singularities. Let $m$ be the ADM mass and $m_{\operatorname{ZAS}}(\Sigma)$ the ZAS mass.

Mixed Riemannian ZAS inequality conjecture. The ADM mass satisfies

$$m\geq \sqrt{\frac{|S|_g}{16\pi}}+m_{\operatorname{ZAS}}(\Sigma).$$

Here $S$ and $\Sigma$ need not be connected, and $m_{\operatorname{ZAS}}(\Sigma)$ is nonpositive.

The conjecture combines the Riemannian Penrose and ZAS inequalities when black holes and zero area singularities coexist. The source expects equality only for flat, Schwarzschild, and Schwarzschild ZAS metrics, possibly with points deleted.