  We show that, for any number of components, the group of braids up to
link-homotopy is torsion-free. This generalizes a result of Humphries up to six
components, and provides an explicit solution to a question posed by Lin and
addressed by Linell and Schick regarding the existence of non-abelian
torsion-free quotients of the braid group. The proof relies on the diagrammatic
theory of welded braids and uses the Artin representation. As a corollary, we
obtain yet another proof that braid groups themselves are torsion-free.
