The strong virtual Legendrian isotopy conjecture for sphere fibers

A strong allowable virtual Legendrian isotopy is a virtual Legendrian isotopy during which the universal cover of the underlying manifold $M$ is never homeomorphic to a CROSS, where CROSS means a compact rank-one symmetric space. A sphere fiber is a fiber of the spherical cotangent bundle $ST^*M$.

Strong virtual Legendrian isotopy conjecture. There exists no strong allowable positive, or non-constant non-negative, virtual Legendrian isotopy of the sphere fiber of $ST^*M$ to itself.

This conjecture extends the known obstruction for non-negative Legendrian isotopies of a fiber to itself: such isotopies force the universal cover to have the cohomological properties of a compact rank-one symmetric space. A stronger result is known for positive isotopies, while the stated virtual version remains open.