Kebekus–Kovács pair version of the Viehweg hyperbolicity conjecture

Let $f\colon (U,D)\to V$ be a log smooth family with $V$ quasi-projective, with all coefficients of $D$ in $(0,1)$, and suppose that every fiber $(U_y,D_y)$ is of log general type. Write $\kappa(V)$ for the Kodaira dimension of $V$ and $\operatorname{Var}(f)$ for the variation of the family. Kebekus–Kovács pair conjecture. Either

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

or

$$\operatorname{Var}(f)\leq\kappa(V).$$

This conjecture generalizes the Viehweg hyperbolicity conjecture to families with arbitrary variation, in a logarithmic pair setting. The first alternative is weaker than maximal variation, while the paper’s main theorem provides evidence for the proposed statement.