The Hawking mass bound for stable marginally outer trapped surfaces

Let $\Sigma$ be a two-surface with area $A$ and mean curvature $H$, and define its Hawking mass by

$$m_H(\Sigma)=\sqrt{\frac{A}{16\pi}}\left(1-\frac{1}{16\pi}\int_{\Sigma}H^2\,d\sigma\right).$$

Suppose that $\Sigma$ is a stable marginally outer trapped surface (MOTS) with outgoing expansion $\theta^+=0$. Hawking mass bound conjecture. One should have

$$m_H(\Sigma)\geq 0,$$

with equality for minimal surfaces in flat space. A complete proof of the surrounding quasi-local mass inequalities is stated to remain open.