The charged Riemannian Penrose inequality for Einstein-Maxwell-Dilaton black holes

An Einstein-Maxwell-Dilaton (EMD) initial data set consists here of a strongly asymptotically flat $(M,g,E,B,\phi)$ with outermost minimal surface boundary of area $A=4\pi\rho^2$, ADM mass $m$, and total charge $q$, satisfying the EMD dominant energy condition. The charge densities are $\operatorname{div}(e^{-2\phi}E)$ and $\operatorname{div}B$.

EMD charged Riemannian Penrose inequality. If the charge densities $\operatorname{div}(e^{-2\phi}E)$ and $\operatorname{div}B$ are compactly supported, then

$$m\geq \frac12\sqrt{\rho^2+2q^2}.$$

If $e^{-2\phi}E$ and $B$ are divergence free, then equality holds if and only if the data set arises as the canonical slice of the Gibbons EMD black hole.

This is the conjectured EMD analogue of the charged Riemannian Penrose inequality proved in the Einstein-Abelian-Yang-Mills setting. The proposed equality characterization is part of the conjecture; the surrounding discussion suggests that a suitable EMD charged Hawking mass and conformal flow would establish it under the relevant horizon and charge assumptions.