The $C^0$ Strong Cosmic Censorship conjecture for charged scalar field black holes

Let $g$ be the maximal globally hyperbolic development of generic regular data for the Einstein–Maxwell–Klein–Gordon system

$$\begin{aligned} \operatorname{Ric}_{\mu\nu}(g)-\frac12R(g)g_{\mu\nu}&=\mathbb{T}^{EM}_{\mu\nu}+\mathbb{T}^{KG}_{\mu\nu},\\ \nabla^\mu F_{\mu\nu}&=q_0\frac{i(\phi\overline{D_\nu\phi}-\overline\phi D_\nu\phi)}2,\qquad F=dA,\\ g^{\mu\nu}D_\mu D_\nu\phi&=m^2\phi, \end{aligned}$$

where $D=\nabla+iq_0A$. $C^0$ Strong Cosmic Censorship conjecture. The maximal globally hyperbolic development is inextendible as a continuous Lorentzian manifold. The conjecture is false: dynamical charged and massive charged scalar-field black holes admit continuous extensions under the assumptions described in the paper.