Pre-compactness of globally hyperbolic spacetimes with Ricci lower bounds

Let $(M,h)$ be an $n$-dimensional Riemannian manifold, and let $g$ be a Lorentzian metric such that $(M,g)$ is a globally hyperbolic spacetime. Write $\operatorname{ric}^{(M,g)}\geq K$ for the stated lower bound on the Ricci tensor, and let $\ell$ denote the convergence used for Lorentzian spaces. Pre-compactness conjecture. The class

$$\left\{(M,h,g): (M,g)\text{ is a globally hyperbolic spacetime},\ \operatorname{ric}^{(M,g)}\geq K\right\}$$

is pre-compact with respect to $\ell$-convergence. This conjecture proposes extending the preceding pre-compactness theorem from smooth generalized cones to globally hyperbolic spacetimes under a full Ricci-tensor lower bound. Its resolution and the precise role of the auxiliary Riemannian metric $h$ are not supplied here.