Arakelov equality characterization for semistable families of varieties

Let $f:X\to Y$ be a semistable non-birationally isotrivial family of $n$-folds over a smooth projective curve $Y$ with discriminant locus $S$. For a vector bundle $W$ on $Y$, write $\mu(W)=\deg W/\operatorname{rank} W$. Assume that the Arakelov equalities hold:

$$\mu(f_*\omega_{X/Y}^\nu)=\frac{n\nu}{2}\cdot\deg\Omega^1_Y(\log S),\qquad \forall\,\nu\in\mathbb N\text{ with }f_*\omega_{X/Y}^\nu\neq 0.$$

Arakelov equality characterization. The general fiber has Kodaira dimension zero, and the family is Shimura in the following sense: $Y$ parameterizes a compactified universal family of abelian varieties $f':X'\to Y$ with the given Mumford–Tate group, and the variation of Hodge structure on the middle cohomology of $f:X\to Y$ is a direct factor of a tensor product of the weight-$1$ variation of Hodge structure attached to the universal family $f':X'\to Y$.

This conjecture seeks to characterize families attaining equality in all nonzero pluricanonical Arakelov slope bounds. It is motivated by the known characterization of Arakelov equality for semistable families of abelian varieties and by the Coleman–Oort conjecture concerning Shimura subvarieties in the Torelli locus.