Specialness conjecture for bi-Hom schemes

Let $M$) be a moduli space satisfying the conditions in Theorem~. A maximal non-rigid family is a product family $H_1\times H_2\to M$ obtained from the bi-Hom construction and maximal with respect to this property.

Specialness conjecture. For any maximal non-rigid family $H_1\times H_2\to M$, both $H_i$ are weakly special subvarieties with respect to the $\mathbb Z$-PVHS of the universal family.

This conjecture is presented as an analogue of Moonen's characterization of special subvarieties of Shimura varieties. Its resolution is not specified in the source.