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.