  In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the
Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by
combining their discoveries with the work of Schiffmann-Vasserot on the
symmetric function side and the work of Hikita and Gorsky-Mazin on the
combinatorial side, were led to formulate an infinite family of conjectures
that extend the original Shuffle Conjecture of Haglund et al. In fact, they
formulated one conjecture for each pair (m,n) of coprime integers. This work of
Gorsky-Negut leads naturally to the question as to where the Compositional
Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent
developments. Our discovery here is that there is a compositional extension of
the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime
and k > 1.
