Mod $p$ geometric André–Oort conjecture for factor-separating subvarieties

Let $\mathbf{I}$ be a finite index set, let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ separate factors over $\mathscr{A}_{g_{\mathbf I},\mathbb{F}}$, and for each $i\in\mathbf I$ let $\mathscr{X}_{f_i}$ be the naïve integral model of an element of $\mathrm{MS}_{f_i}(\mathcal{A}_{g_i})$. Suppose that $X$ contains a Zariski dense collection $\Xi$ of positive-dimensional special subvarieties. Mod $p$ geometric André–Oort conjecture for factor-separating subvarieties. There exists $i_0\in\mathbf I$ such that the Zariski closure of $X$ in $\mathscr{A}_{g_{\mathbf I},\mathbb{F}}^{\mathrm{ord}}$ is the product of a component of $\mathscr{X}_{f_{i_0},\mathbb{F}}^{\mathrm{ord}}$ with a subvariety $Y\subseteq\prod_{i\in\mathbf I\setminus\{i_0\}}\mathscr{X}_{f_i,\mathbb{F}}^{\mathrm{ord}}$. This is the factor-separated variant used to derive the geometric André–Oort conclusion; the source gives no resolution status.