Cauchy–BTZ extension maximality conjecture for globally hyperbolic singular manifolds

Let $M_0$ be a globally hyperbolic singular manifold. Let $M_1$ be its maximal Cauchy-extension, let $M_2$ be the maximal BTZ-extension of $M_1$, and let $M_3$ be the maximal Cauchy-extension of $M_2$.

Cauchy–BTZ extension maximality conjecture. The manifold $M_3$ is both Cauchy-maximal and BTZ-maximal.

The preceding example shows that taking a maximal Cauchy-extension followed by a maximal BTZ-extension need not produce a Cauchy-maximal manifold. The conjecture asserts that one further maximal Cauchy-extension produces a manifold maximal with respect to both types of extension.