Generalized Riemannian Penrose inequality with charge and cosmological constant

Let $(M,\gamma,E,\Lambda)$ be an electrically charged Riemannian manifold of dimension $n+1\geq3$ which is geodesically complete up to its closed, strictly outward minimizing minimal boundary $\partial M$. Assume that $(M,\gamma,E,\Lambda)$ is asymptotically Euclidean if $\Lambda=0$ and asymptotically hyperbolic with asymptotic hyperbolic radius $\sqrt{-\tfrac{n(n+1)}{2\Lambda}}$ if $\Lambda<0$, respectively, and denote its mass and charge by $m$ and $q$. If $q\neq0$, assume in addition that $\partial M$ is connected. Furthermore, assume that the Dominant Energy Condition holds on $M$. Generalized Riemannian Penrose inequality with charge. Then

$$\frac{1}{2}\left[\left(\frac{|\partial M|_{g}}{\omega_n}\right)^{\frac{n-1}{n}}+q^2\left(\frac{|\partial M|_{g}}{\omega_n}\right)^{-\frac{n-1}{n}}-\frac{2\Lambda}{n(n+1)}\left(\frac{|\partial M|_{g}}{\omega_n}\right)^{\frac{n+1}{n}}\right]\leq m,$$

where $|\partial M|_{g}$ denotes the volume of $\partial M$ with respect to the induced metric $g$ on $\partial M$, and $\omega_n=|\mathbb{S}^{n}|_{g_*}$ denotes the volume of the unit round sphere $(\mathbb{S}^{n},g_*)$. Equality holds if and only if $(M,\gamma,E,\Lambda)$ is isometric to the necessarily sub-extremal $\Lambda$-Reissner--Nordström manifold $(\overline{M_{m,q,\Lambda}},\gamma_{m,q,\Lambda},E_{m,q,\Lambda},\Lambda)$ of mass $m$, charge $q$, and cosmological constant $\Lambda$. This conjectured inequality refines the charged Riemannian Penrose inequality by incorporating a nonpositive cosmological constant. The relevant mass--charge inequality is established for $\Lambda=0$, while its extension to $\Lambda<0$ was not known in the source; the generalized Penrose inequality and its rigidity statement are presented as conjectural.