The higher-dimensional Ricci compactness conjecture

Let $M$ be complete, let $O$ be a fixed origin, and suppose that for some $r_0>0$ the Ricci curvature is bounded below by $(n-1)/(4r^2)$ whenever $r\geq r_0$. Higher-dimensional Ricci compactness conjecture. If there are $k$ linearly independent geodesics through $O$ along which

$$\operatorname{Ric}\geq\frac{n-1}{r^2}\left(\frac14+\nu^2\right),$$

then $M$ is compact. The source presents this as a proposed compactness criterion and gives no resolution.