We prove that closed connected contact manifolds of dimension $geq 5$ related by an h-cobordism with a flexible Weinstein structure become contactomorphic after some kind of stabilization. We also provide examples of non-conjugate contact structures
on a closed manifold with exact symplectomorphic symplectizations.
We provide examples of contact manifolds of any odd dimension $geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations.
