Weak positive-Legendrian-isotopy CROSS conjecture

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, and let $\phi_t$ be a positive Legendrian isotopy of $(S^{\ast}M,\xi)$ with $\phi_0=\operatorname{id}$. For $x,y\in M$, assume that there is $T>0$ such that every $v\in S^{\ast}_xM$ satisfies $\phi_t(v)\in S^{\ast}_yM$ for some $t\in(0,T]$. Weak positive-Legendrian-isotopy 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 states that the compactness and fundamental-group conclusions are proved for isotopies arising from the specified null-geodesic construction, while the general conjecture remains open.