Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base $Y^{\circ }$, and suppose the family is non-isotrivial. If $Y$ is a smooth compactification of $Y^{\circ }$, such that $D:=Y\setminus Y^{\circ }$ is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along $D$. Viehweg and Zuo have shown that for some $m\>0$, the $m^{\mathrm{th}}$ symmetric power of this sheaf admits many sections. More precisely, the $m^{\mathrm{th}}$ symmetric power contains an invertible...