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Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field 
$K$ and a positive integer 
$g$, there is an integer 
$m_{0}$ such that for any 
$m>m_{0}$ there is no principally polarized abelian variety 
$A/K$ of dimension 
$g$ with full level-
$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.
