Quantum very strong cosmic censorship conjecture

Let $(M,g_{ab})$, $S$, $D(S)$, $\mathscr{A}$, $\mathscr{B}$ and $\omega$ be as in the quantum strong cosmic censorship conjecture, and suppose that $(D(S),\omega)$ satisfies the semiclassical gravity equations. Quantum very strong cosmic censorship conjecture. Generically, there exists no extension of $\omega$ to a state $\overline\omega:\mathscr{A}(M;\overline{D(S)})\to\mathbb{C}$ such that $\overline\omega-H\in L^1_{\mathrm{loc}}$ for a Hadamard parametrix $H$ on $\overline{D(S)}$; equivalently, $\overline\omega-H$ cannot be defined as a distribution on $\overline{D(S)}$. This strengthens the preceding claim by ruling out extensions that differ from a Hadamard state only by a locally integrable regular term at the Cauchy horizon; it is open.