  Greg Hjorth and Simon Thomas proved that the classification problem for
torsion-free abelian groups of finite rank \emph{strictly increases} in
complexity with the rank. Subsequently, Thomas proved that the complexity of
the classification problems for $p$-local torsion-free abelian groups of fixed
rank $n$ are \emph{pairwise incomparable} as $p$ varies. We prove that if
$3\leq m<n$ and $p,q$ are distinct primes, then the complexity of the
classification problem for $p$-local torsion-free abelian groups of rank $m$ is
again incomparable with that for $q$-local torsion-free abelian groups of rank
$n$.
