Borde–Sorkin conjecture on causal continuity of Morse spacetimes

Let a Morse spacetime be a singular Lorentzian spacetime built from a Morse function, whose critical points have an index and a coindex. Borde–Sorkin conjecture. A Morse spacetime is causally continuous if and only if all critical points of the Morse function have index and coindex different from $1$.

The conjecture asserts that causal discontinuity occurs only when the stable or unstable manifold of a critical point is disconnected. It is known for a larger family of Morse spacetimes under a near-isotropy assumption on the Morse-function coefficients, but remains unresolved in general.