The characteristic-profile scattering conjecture for the Einstein–Maxwell–Klein–Gordon system

Let $\phi_{\mathcal H^+}$ be a suitable finite-energy scalar-field profile on the event horizon of a spherically symmetric Einstein–Maxwell–Klein–Gordon black hole, and let $\Sigma$ be an asymptotically flat Cauchy hypersurface. Characteristic-profile scattering conjecture. There exist sufficiently regular Cauchy data on $\Sigma$ for the spherically symmetric Einstein–Maxwell–Klein–Gordon system that generate a dynamical black hole whose scalar field along the event horizon is precisely $\phi_{\mathcal H^+}$. If true, this would show that a broad class of horizon profiles—including profiles lacking the oscillation condition—can arise from Cauchy evolution, and would clarify the distinction between generic and fine-tuned data. The conjecture is presented on the basis of scattering arguments and is not proved here.