In any closed contact manifold of dimension at least , we construct examples of closed Legendrian submanifolds which are not diffeomorphic but whose Lagrangian cylinders in the symplectization are Hamiltonian isotopic.
"Legendrian submanifolds with Hamiltonian isotopic symplectizations." Algebr. Geom. Topol. 16 (6) 3641 - 3652, 2016. https://doi.org/10.2140/agt.2016.16.3641