The characteristic $p$ André–Oort conjecture for products of GSpin Shimura varieties

Let $f:X_0\to\mathscr{S}_{\mathbf{I},F_{\infty}}^{\operatorname{ord}}$ be a locally closed immersion, and let $X$ be its image. Suppose that $X$ possesses a Zariski-dense collection $\A$ of positive-dimensional special subvarieties. Define $\I_{\A}$ to be the set of indices whose projections contain a Zariski-dense collection of special subvarieties with positive-dimensional projections. For indices in $\I_{\A}$, let $X_i$ be the corresponding projections, and decompose their Shimura data into simple adjoint factors indexed by $\J$; let $\J_{\A}$ record the factors with positive-dimensional projected special subvarieties. The characteristic $p$ André–Oort conjecture for products of GSpin Shimura varieties. The variety $X$ is quasi-weakly special; if $\#\mathbf{I}=1$, it is special. More precisely: (1) each $X_i$ for $i\in\mathbf{I}_{\mathbf{A}}$ is special; (2) $X$ is the product of a quasi-weakly special subvariety $Y\subseteq\mathscr{S}_{\mathbf{I}_{\mathbf{A}},F_{\infty}}$ with a subvariety of $\mathscr{S}_{\mathbf{I}-\mathbf{I}_{\mathbf{A}},F_{\infty}}$; and (3) $Y$ is an almost product of a special subvariety of $\prod_{j\in\mathbf{J}_{\mathbf{A}}}\mathscr{Y}_{j,F_{\infty}}^{\operatorname{ord}}$ with a subvariety of $\prod_{j\in\mathbf{J}-\mathbf{J}_{\mathbf{A}}}\mathscr{Y}_{j,F_{\infty}}^{\operatorname{ord}}$. In particular, if each $X_i$ is simple, then $Y$ is special.

This is the product form of the characteristic-$p$ André–Oort prediction, refining quasi-weak specialness by an almost-product description. The supplied passage does not give a resolution status.