Weak contact-geometric CROSS conjecture for Reeb flows

Let $M$ be a manifold of dimension at least $2$. Let $(S^{\ast}M,\xi)$ be its spherical cotangent bundle with the standard contact structure, let $\alpha$ be a contact form with $\xi=\ker(\alpha)$, and let $\varphi_{\alpha}^t$ be its Reeb flow. For $x,y\in M$, assume that there is $T>0$ such that every $v\in S^{\ast}_xM$ reaches $S^{\ast}_yM$ at some $t\in(0,T]$. Weak contact conjecture. Then $M$ is compact, its fundamental group is finite, and the integral cohomology ring of its universal cover is that of a CROSS. The source distinguishes this bounded-time version from the strong conjecture and reports the compactness and fundamental-group conclusions in the Riemannian case; the general statement remains open.