Zannier's conjecture on torsion points in the universal abelian family

Let $\pi: \mathfrak{A}_{g,l} \to A_{g,l}$ be the universal family of complex principally polarized abelian varieties of dimension $g$ with symplectic level $l$-structure, and let $A_0$ be a fixed complex abelian variety. Let $\mathcal{V} \subset \mathfrak{A}_{g,l}$ be an irreducible subvariety containing a Zariski dense set of points $p \in \mathfrak{A}_{g,l}(\mathbb{C})$ such that the fiber $(\mathfrak{A}_{g,l})_{\pi(p)}$ is isogenous to $A_0$ and $p$ is torsion on $(\mathfrak{A}_{g,l})_{\pi(p)}$. Zannier's conjecture. Then $\pi(\mathcal{V})$ is a totally geodesic subvariety of $A_{g,l}$, and $\mathcal{V}$ is an irreducible component of a subgroup scheme of

$$\mathfrak{A}_{g,l} \times_{A_{g,l}} \pi(\mathcal{V}) \to \pi(\mathcal{V}).$$

Gao has proved the conjecture when $\dim \pi(\mathcal{V}) \leq 1$; the general statement is therefore resolved.