  In the early 2000's the first and second named authors worked for a period of
six years in an attempt of proving the Compositional Shuffle Conjecture [1].
Their approach was based on the discovery that all the Combinatorial properties
predicted by the Compositional Shuffle Conjecture remain valid for each family
of Parking Functions with prescribed diagonal cars. The validity of this
property was reduced to the proof of a functional equation satisfied by a
Catalan family of univariate polynomials. The main result in this paper is a
proof of this functional equation. The Compositional Shuffle Conjecture was
proved in 2015 by Eric Carlsson and Anton Mellit [3]. Our proof of the
Functional Equation removes one of the main obstacles in the completion of the
Garsia-Hicks approach to the proof of the Compositional Shuffle Conjecture. At
the end of this writing we formulate a few further conjectures including what
remains to be proved to complete this approach.
