  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 $A/K$ 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 $m > m_X$.
