Conjecture on the space-like character of the singularity

Let $\mathcal{Q}^+$ be the quotient spacetime, $\mathcal{I}^+$ future null infinity, and $\mathcal{S}$ the relevant singular boundary component. Space-like singularity conjecture. Among all admissible data, there exists a generic subclass such that, if $\mathcal{Q}^+\cap J^-(\mathcal{I}^+)\neq\emptyset$, then $\mathcal{S}\neq\emptyset$ and $\mathcal{S}$ is space-like in the sense of the paper's definition. The paper explicitly says that its method does not control the causal character of $\mathcal{S}$, so this conjecture remains open.