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We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q,t-enumerates, by the statistics dinv and area, the parking functions whosesupporting Dyck path touches the main diagonal according to the composition pof size a + b + c and have a reading word which is a shuffle of one decreasingword and two increasing words of respective sizes a, b, c. Here Cp(1) is arescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperatorintroduced in [1]. This is our latest progress in a continued effort to settlethe decade old shuffle conjecture of [14]. It includes as special cases allprevious results connected with this conjecture such as the q, t-Catalan [3]and the Schroder and h, h results of Haglund in [12] as well as theircompositional refinements recently obtained in [9] and [10]. It also confirmsthe possibility that the approach adopted in [9] and [10] has the potential toyield a resolution of the shuffle parking function conjecture as well as itscompositional refinement more recently proposed by Haglund, Morse and Zabrockiin [15].

A three shuffle case of the compositional parking function conjecture