I'm interested in understanding better the properties of the integrable (almost) complex structure that Eliashberg constructs on the cotangent bundle of a closed manifold $M$, whose dimension is at least $3$ (See these notes, thm 11.10).
Let $\omega_{\text{can}}$ the canonical symplectic structure on the cotangent bundle, and let $J_{\text{can}}$ a tamed a.c.s.
The statement of Theorem 11.10 says (in our setting) that given $(T^*M, \omega_{\text{can}}, J_{\text{can}}) $ we can find an exhausting Morse function $\phi$ with some properties, such that $(T^*M, \omega_{\phi}, \widetilde{J})$ is a Stein manifold. Moreover $\widetilde{J}$ is homotopic to $J_{\text{can}}$ and $\widetilde{J}$ is tamed by $\omega_{\phi}$.
The notation $\omega_{\phi}$ should suggest that the new symplectic structure is build from the map $\phi$.
I'd like to know if we can assume that $\widetilde{J}$ is tamed by $\omega_{\text{can}}$, or what kind of relation we can force them to have.
In this answer, T. Perutz concludes by saying that it seems to be the case that $\omega_{\phi}$ and $\omega_{\text{can}}$ are symplectomorphic (apparently thanks to a clever choice for $\phi$).
I'd like to see some references for this statement, since I wasn't able to find such a symplectomorphism between the two structures anywhere in the literature.
