Constructing holomorphic functions on universal coverings of complex algebraic varieties

What complex analytic spaces can be obtained as the universal covering of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal covering of any smooth projective variety X is necessarily holomorphically convex.
In other words, is there a proper holomorphic map from the universal covering of X to a Stein analytic space? Although still open, Shafarevich's question has received partial positive answers, for example when the fundamental group of X admits a faithful complex linear representation (Eyssidieux-Katzarkov-Pantev-Ramachandran). In my talk, I will discuss the generalization of Shafarevich's question to non-compact algebraic varieties. This is joint work with Ben Bakker and Jacob Tsimerman.