Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture

Assuming Lang’s conjecture, we prove that for a fixed prime $p$, number field $K$, and positive integer $g$, there is an integer $r$ such that no principally polarized abelian variety $A\mathrm{/}K$ of dimension $g$ has full level $p^r$ structure. To this end, we use a result of Zuo to prove that for each closed subvariety $X$ in the moduli space ${\mathcal{A}}_g$ of principally polarized abelian varieties of dimension $g$, there exists a level $m_X$ such that the irreducible components of the preimage of $X$ in ${\mathcal{A}}_g^{[m]}$ are of general type for $m>m_X$.