A classical conjecture of Shafarevich, solved by Parshin and Arakelov, predicts that any smooth projective family of high genus curves over the complex line minus a point or an elliptic curve is isotrivial (has zero variation in its algebraic structure). A natural question then arises as to what other families of manifolds and base spaces might behave in a similar way. Kebekus and Kov\'acs conjecture that families of manifolds with good minimal models form the most natural category where Shafarevich-type conjecturesshould hold. For example, analogous to the original setting of Shafarevich Conjecture, they expect that over a base space of Kodaira dimension zero such families are always (birationally) isotrivial. In this talk I will discuss a solution to Kebekus-Kov\'acs Conjecture.