The H¹ strong cosmic censorship conjecture for the Einstein-Maxwell-charged-Klein-Gordon system

Let $\Sigma$ be a regular, compact spacelike hypersurface without boundary. Prescribe generic initial data in the spherically symmetric class for the Einstein-Maxwell-charged-Klein-Gordon system with $\Lambda>0$, and let its future maximal globally hyperbolic development be the largest Lorentzian development determined by these data. $H^1_{\mathrm{loc}}$ strong cosmic censorship conjecture. This development cannot be locally extended as a Lorentzian manifold endowed with a continuous metric and Christoffel symbols in $H^1_{\mathrm{loc}}$. This is a stronger regularity formulation than mere continuous inextendibility and is motivated by the regularity needed to interpret extensions as weak solutions of the Einstein equations. The source does not establish the conjecture as a theorem.