Let $X$ be a complex manifold endowed with a holomorphic closed 2-form $\omega$ whose associated map $\omega : TX \to T^*X$ is invertible. Can we always embed $X$ as an open subset of a compact complex manifold $Y$ endowed with a holomorphic Poisson structure $\pi : T^*Y \to TY$ such that $\pi|_X = \omega^{-1}$? We may assume that $X$ is quasi-projective, or other reasonable assumptions, if that helps.

Note that this holds, for example, if $X$ is the cotangent bundle of a compact manifold.