Compactness conjecture for manifolds satisfying a generalized curvature-dimension inequality

Let $\mathbb M$ be the contact manifold considered in the paper, with curvature-dimension parameters $\rho_1$, $\rho_2$, $\rho_3$ and $\kappa$. Compactness conjecture. If

$$\rho_1-\frac{\kappa\sqrt{\rho_3}}{\sqrt{\rho_2}}>0,$$

then the manifold $\mathbb M$ is compact. The conjecture concerns the optimality of the paper's compactness theorem, which derives compactness under stronger geometric conditions via the Bonnet–Myers theorem for a suitably rescaled Riemannian metric; its resolution is not established in the supplied text.