Kobayashi's conjecture on fundamental groups of positively curved semi-Riemannian manifolds

Let $n$ and $q$ be positive integers with $n \geq 2q$. Let $(M,g)$ be an $n$-dimensional geodesically complete semi-Riemannian manifold of index $q$, and suppose its sectional curvature has a positive lower bound. Kobayashi's conjecture. The manifold $M$ is never compact, and, if $n \geq 3$, its fundamental group $\pi_1(M)$ is finite. The conjecture is presented as a perturbative analogue of the Calabi–Markus–Wolf finiteness theorem for positive constant curvature; the supplied text also says that it is true by Kulkarni's theorem, which reduces the one-sided sectional-curvature bound to constant curvature.