Specialness conjecture for maximally non-rigid morphisms

Let $\mathcal M_h$ be a moduli space satisfying the conditions in Theorem~, and let

$$H_1\times H_2\to\mathcal M_h$$

be the maximally non-rigid morphism constructed in Proposition~.

Specialness conjecture. Both $H_i$ are weakly special subvarieties with respect to the $\mathbb Z$-PVHS associated with the universal family.

The conjecture is motivated by Saito's classification and Viehweg–Zuo's example. Its resolution is not specified in the source.