Quantum strong cosmic censorship without F-locality

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, with $\hat g_{ab}=\psi^{-1*}g_{ab}$. Let $\mathscr{B}$ be a free or perturbatively interacting quantum field theory over $(D(S),\hat g_{ab})$, and let $\omega:\mathscr{B}\to\mathbb{C}$ be a pure Hadamard state with two-point function $\omega_2\in\mathscr{D}'(D(S)\times D(S))$. Quantum strong cosmic censorship without $F$-locality. The extension of $\omega_2$ as a distribution in $\mathscr{D}'(\overline{D(S)}\times\overline{D(S)})$ fails to satisfy the Hadamard condition at the boundary. This formulation removes the $F$-locality assumption and directly constrains the boundary behaviour of the two-point function; it remains open.