Viehweg's conjecture on maximal variation and log general type

Let $f^\circ:X^\circ\to S^\circ$ be a smooth family of canonically polarized varieties. The family has maximal variation when its variation equals the dimension of the base. Viehweg's conjecture. If $f^\circ$ is of maximal variation, then $S^\circ$ is of log general type. This conjecture predicts that maximal variation forces strong logarithmic positivity of the base; the paper gives an affirmative answer for families parametrized by surfaces.