Char $p$ geometric André–Oort conjecture

Let $f:X\rightarrow\mathscr{A}_{g,\mathbb{F}}$ be a locally closed immersion with image in the ordinary locus. A subvariety is quasi-weakly special when its ordinary locus is contained in a special subvariety splitting as an almost product with a positive-dimensional factor, and its closure splits correspondingly. Char $p$ geometric André–Oort conjecture. Exactly one of the following holds: $X$ is quasi-weakly special, or $X$ does not contain a Zariski dense collection of positive-dimensional special subvarieties. This is the geometric André–Oort alternative in characteristic $p$; the source later proves it under the Mumford–Tate hypothesis, but gives no unconditional resolution here.