Finiteness conjecture for globally hyperbolic Lorentzian manifolds

Let $M$ be a non-spacelike geodesically complete globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface of dimension $m \geqslant 3$. Suppose there is a positive constant $k$ such that $M$ satisfies the curvature condition $R \geqslant k$, meaning that

$$\langle R(u,v)v,u\rangle \geqslant k\bigl(\langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2\bigr)$$

for all tangent vectors $u,v$. Finiteness conjecture. The fundamental group of $M$ is finite. This is presented as a reformulation of Kobayashi's conjecture for the Lorentzian setting, designed to include Lorentzian manifolds of variable curvature. The supplied text does not explicitly state whether this reformulated claim has been proved, so its status remains open here.