Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$? 


Remark. I am aware of examples (due to Mike Davis) of compact $n$-manifolds whose universal covering spaces are fake $R^n$'s i.e. are contractile but not homeomorphic to $R^n$. 



Exotic $R^4$ as the universal covering space
This is problem 4.79(A) of Kirby's 1995 problem list, contributed by Gompf. It is my impression that it is still open. 


There is some small progress: Remark 7.2 in [this article](https://arxiv.org/pdf/math/9712292.pdf) observes that their constructions imply that a specific countable set of examples of exotic $\Bbb R^4$s cannot possibly cover a closed manifold. This is not a huge reduction, as only countably many exotic $\Bbb R^4$s could possibly be covers of the countable set of closed smooth 4-manifolds, anyway. But at least the examples are somewhat explicit. 


I couldn't find any other references to this problem in the literature, but that doesn't mean there aren't any. 


UPDATE: I emailed Bob Gompf; he is not aware of any recent progress. 


