The virtual Legendrian isotopy conjecture for spheres and complex projective spaces

Let $S^m$ be the $m$-sphere, let $\mathbb C P^n$ be complex projective $n$-space, and let $J^1N$ denote the 1-jet space of a manifold $N$. A virtual Legendrian isotopy is an isotopy allowing the virtual Legendrian modifications considered in the paper, while a Legendrian isotopy is an ordinary isotopy through Legendrian links.

Virtual Legendrian isotopy conjecture. Two Legendrian links in $ST^*S^m$ or $ST^*\mathbb C P^n$ are Legendrian isotopic if and only if they are virtually Legendrian isotopic. The same assertion is made for Legendrian links in $J^1S^m$ and $J^1\mathbb C P^n$. More generally, an analogous statement is proposed for fibers of $ST^*M$ when $M$ is a CROSS.

The sphere case in dimension two follows from the surface result cited in the paper, but the higher-dimensional cases and the corresponding statements for complex projective spaces and general CROSSes are left open. The sphere case was first formulated by the first author in the cited work of Cahn and Levi.