Viehweg's induced inequality for relative Kodaira dimension

Let $f:X\to S$ be a fibre space $X/S$, let $X_{\bar{\eta}}$ be its generic geometric fibre, and let $\omega_{X/S}$ and $\omega_{X_{\bar{\eta}}}$ denote the relative and generic-fibre canonical sheaves. Let $\operatorname{var}(X/S)$ denote the variation of $X/S$. Viehweg's induced inequality. The conjecture implies

$$\kappa(\omega_{X/S})\geq \kappa(\omega_{X_{\bar{\eta}}})+\operatorname{var}(X/S).$$

This is presented as a consequence of Viehweg's conjecture rather than as an independent conjecture. Its status therefore follows from the status of the preceding conjecture.