Good compact 4-dimensional complex/symplectic/Kähler orbifolds that are not very good

Determine whether there exists a good compact orbifold of real dimension 4 carrying an almost complex structure, a symplectic structure, or a Kähler structure, that is not very good (i.e., admits a covering by a manifold but no finite manifold cover).

Background

Extending the inquiry to geometric structures beyond contact, the authors ask for dimension-4 examples within complex categories—almost complex, symplectic, and Kähler—that are good but not very good.

They note a potential route to complex orbifold examples in complex dimension 3 as twistor spaces obtained from their 4-dimensional constructions, contingent on an orbifold version of Taubes’s result, but the dimension-4 existence question remains explicitly posed.