Variation of singular K\"ahler-Einstein metrics: positive Kodaira dimension

Given a K"ahler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular K"ahler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X\mathrm{/}Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted K"ahler-Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiber-wise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).