  The Dyck path algebra construction of Carlsson and Mellit from
arXiv:1508.06239 is interpreted as a representation of "the positive part" of
the group of toric braids. Then certain sums over $(m,n)$-parking functions are
related to evaluations of this representation on some special braids. The
compositional $(km,kn)$-shuffle conjecture of Bergeron, Garsia, Leven and Xin
from arXiv:1404.4616 is then shown to be a corollary of this relation.
