Let $\Sigma$ be a closed, orientable surface.

Then the cotangent bundle $T^*\Sigma$ has a canonical symplectic form $\omega$, given as the derivative of the tautological Liouville one-form. We can modify it to a "magnetic" form by adding some two-form $\sigma$ on the base to the symplectic form.

The notation $T^*_\sigma \Sigma$ will denote the "magnetic cotangent bundle", i.e. the cotangent bundle equipped with a symplectic form $\omega + \sigma$.

Given this, my (rather broad) question is the following: in what cases (i.e. varying $\sigma$ or the genus of $\Sigma$) is it known that a small neighborhood of the zero section in $T^*_\sigma \Sigma$ symplectically embeds into a closed symplectic $4$-manifold?

For the purposes of this question, we will suppose that $\sigma \neq 0$. Otherwise, there are many examples with $\sigma = 0$, as by the Weinstein neighborhood theorem, we can just take a neighborhood of an embedded Lagrangian $\Sigma$ in a closed symplectic $4$-manifold.