Let $M = B^2_R \times S^1$ where $B^2_R$ is the 2-dimensional open ball with radius $R (>1)$. Take two contact structures $\alpha_1 = \frac{1}{2} (xdy - ydx) + dt$ and $\alpha_2 = \frac{1}{2} (xdy - ydx) + 2 dt$.
Question: is there a diffeomorphism $\Phi$ on $M$ such that $\Phi^*\alpha_2= e^g \alpha_1$ for some function $g$ on $M$?
The same question can be formulated for $M = \mathbb R^2 \times S^1$. In both cases, non-compactness here stops me applying Gray's theorem directly.