Abstract: A conjecture of Shafarevich and Viehweg, settled through  the works of Viehweg-Zuo, Campana-Paun and Popa-Schnell, predicted that a family of maximally-varying, smooth, projective manifolds with good minimal models have (log-)general type base. Generalizing this problem, Kebekus and Kovács conjecture that the Kodaira dimension of base spaces of such families should define an upper bound for the variation in the family, even when the variation is not maximal. My aim in this talk is to discuss a strategy to solve this conjecture, when dimension of the base is at most 5.