Suppose we have a contact manifold (M,$\xi$) and two associated contact forms $\alpha$ and $\beta$ s.t.$\beta=f\alpha$ with $f>0$. Suppose also that we have two almost complex structures $J_\alpha$ and $J_\beta$ on $\xi$ compatible with/tamed by $d\alpha$ and $d\beta$ respectively. Is it possible to define a symplectic cobordism $\mathbb{R}\times M$ (with a cylindrical almost complex structure) such that above the level $b$, we have the data associated to $\beta$ and below the level $a$, we have the data associated to $\alpha$ for some $b>a$ and such that for any holomorphic cylinder, we have the upper action $\geq$ the lower action?
Thanks for your interest.