A ${\sigma }_2$ Penrose inequality for conformal asymptotically hyperbolic 4-discs

In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the ${\sigma }_2$ curvature has lower bound ${\sigma }_2\ge \frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space ${\mathbb{H}}^4$ when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.