Viehweg's variation conjecture for fibre spaces

Let $f:X\to S$ be a fibre space $X/S$ whose generic geometric fibre has Kodaira dimension at least $0$. Let $\omega_{X/S}$ denote the relative canonical sheaf, and let $\operatorname{var}(X/S)$ denote the variation of the fibre space. Viehweg's conjecture. There exists a number $m$ such that

$$\kappa\left(\det f_*\omega_{X/S}^{\otimes m}\right)\geq \operatorname{var}(X/S).$$

This conjecture relates the positivity of direct images of relative pluricanonical sheaves to the variation of the family. The source attributes it to Viehweg but gives no general resolution status.