Good compact contact 5-orbifolds that are not very good

Determine whether there exists a good compact contact orbifold of real dimension 5 that is not very good (i.e., admits a covering by a manifold but no finite manifold cover).

Background

The authors observe that unit tangent bundles of certain 4-orbifolds furnish examples of good compact contact 7-orbifolds that are not very good, by analyzing their fundamental groups and singularities.

They then pose the analogous existence question in dimension 5, for which no such examples are currently established in the paper.