Kebekus–Kovács hyperbolicity conjecture II

Let $f_U:U\to V$ be a smooth family of projective varieties, and let $(X,D)$ be a smooth compactification of $V$ with $V\cong X\smallsetminus D$. Assume that the geometric general fiber of $f_U$ has a good minimal model. Kebekus–Kovács hyperbolicity conjecture II. If $\kappa(X,D)\geq0$, then

$$\operatorname{Var}(f_U)\leq\kappa(X,D).$$

This is a reformulation of the first conjecture using the solution of Viehweg's hyperbolicity conjecture. In the canonically polarized case it is known in full generality, but the stated general form remains the conjectural formulation considered here.