Quantum strong cosmic censorship conjecture

Let $S$) be a strictly partial Cauchy surface of the (not necessarily globally hyperbolic) spacetime $(M,g_{ab})$, and let $D(S)$ be its domain of dependence. Regard $(D(S),\hat g_{ab})$ as a globally hyperbolic spacetime, where $\hat g_{ab}=\psi^{-1*}g_{ab}$ for an isometric embedding $\psi:D(S)\to\psi(D(S))\subset M$. Let $\mathscr{A}$ be an $F$-local free or perturbatively interacting quantum field theory over $(M,g_{ab})$, and let $\mathscr{B}$ be a corresponding theory over $(D(S),\hat g_{ab})$ isomorphic to $\mathscr{A}(M;D(S))$. Let $\omega:\mathscr{B}\to\mathbb{C}$ be a pure Hadamard state. Quantum strong cosmic censorship conjecture. There exists no extension of $\omega$ to a Hadamard state $\overline\omega:\mathscr{A}(M;\overline{D(S)})\to\mathbb{C}$. The conjecture asserts that pure Hadamard states cannot remain Hadamard when extended across the Cauchy horizon of a strictly partial Cauchy development; its status is open.