The strong virtual Legendrian unlinking conjecture for cotangent links

Let $M^m$ be a manifold with $mgeq 3$ whose universal cover is not homeomorphic to a CROSS, and let $L$ be a two-component Legendrian link in $ST^*M$ admitting an allowable virtual Legendrian isotopy to a pair of fibers. A strong allowable virtual Legendrian isotopy is one that remains in the class for which the universal cover of the underlying manifold is never homeomorphic to a CROSS.

Strong virtual Legendrian unlinking conjecture. There is no non-negative strong allowable virtual Legendrian isotopy of one component of $L$ to the other.

For surfaces, the analogous assertion is proved in the paper. In higher dimensions it is presented as a conjectural obstruction to non-negative isotopies between components of links that become a pair of fibers; it is also used to approach the causality conjecture for allowable generalized spacetimes.