Structural decomposition conjecture for the non-rigid locus

Let $M$ be the moduli space under consideration and let $\mathrm{NRL}(\mathcal M)$ be its non-rigid locus. Suppose its Zariski closure admits a decomposition into subvarieties.

Structural decomposition conjecture.

$$\overline{\mathrm{NRL}(\mathcal M)}^{\mathrm{Zar}}=\bigcup_{i=1}^nP_i\cup\bigcup_{j=1}^mN_j,$$

where each $P_i$ is dominated by a fiber space whose generic fiber is a product variety, and each $N_j$ is birational to a Shimura variety of rank at least $2$.

The source states that this conjecture follows conceptually from the finiteness conjecture and that an analogous theorem is proved assuming that finiteness conjecture.