Kebekus–Kovács hyperbolicity conjecture I

Let $f_U:U\to V$ be a smooth projective family whose general fiber admits a good minimal model. Write $\operatorname{Var}(f_U)$ for the variation of the family. Kebekus–Kovács hyperbolicity conjecture I. Either

$$\kappa(V)=-\infty\quad\text{and}\quad \operatorname{Var}(f_U)<\dim V,$$

or

$$\kappa(V)\geq 0\quad\text{and}\quad \operatorname{Var}(f_U)\leq\kappa(V).$$

The conjecture extends the Viehweg hyperbolicity conjecture beyond maximal variation and was proved for bases of dimension at most five in the paper; the source does not otherwise establish its general status.