All:
I would appreciate any ideas, refs., etc. on the following:

Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the
contact structure, say $\eta$ on $M^3$ lifts to a contact structure $\eta'$ on $X$.

Just hoping to get some information about the relationship between the tightness/overtwistedness and other contact properties of $(X, \eta')$ and the
contact properties of $(M^3, \eta)$ . Can we, e.g.,  directly conclude from the tightness/overtwistedness of $(x, \eta')$ the tightness/overtwistedness of $(M^3 \eta)$; are there any other known relationships between the contact properties of the two spaces? Maybe some properties of Legendrian knots and their twisting?

Thanks.