  We consider the group of pure welded braids (also known as loop braids) up to
(link-)homotopy. The pure welded braid group classically identifies, via the
Artin action, with the group of basis-conjugating automorphisms of the free
group, also known as the McCool group P $\Sigma$ n. It has been shown recently
that its quotient by the homotopy relation identifies with the group hP
$\Sigma$ n of basis-conjugating automorphisms of the reduced free group. In the
present paper, we describe a decomposition of this quotient as an iterated
semi-direct product which allows us to solve the Andreadakis problem for this
group, and to give a presentation by generators and relations. The Andreadakis
equality can be understood, in this context, as a statement about Milnor
invariants; a discussion of this question for classical braids up to homotopy
is also included.
