  The group of basis-conjugating automorphisms of the free group of rank $n$,
also known as the McCool group or the welded braid group $P\Sigma_n$, contains
a much-studied subgroup, called the upper McCool group $P\Sigma_n^+$. Starting
from the cohomology ring of $P\Sigma_n^+$, we find, by means of a Gr\"obner
basis computation, a simple presentation for the infinitesimal Alexander
invariant of this group, from which we determine the resonance varieties and
the Chen ranks of the upper McCool groups. These computations reveal that,
unlike for the pure braid group $P_n$ and the full McCool group $P\Sigma_n$,
the Chen ranks conjecture does not hold for $P\Sigma_n^+$, for any $n\ge 4$.
Consequently, $P\Sigma_n^+$ is not isomorphic to $P_n$ in that range, thus
answering a question of Cohen, Pakianathan, Vershinin, and Wu. We also
determine the scheme structure of the resonance varieties
$\mathcal{R}_1(P\Sigma_n^+)$, and show that these schemes are not reduced for
$n\geq 4$.
