  Let $W$ be an irreducible real reflection group. Armstrong, Reiner, and the
author presented a model for parking functions attached to W and made three
increasingly strong conjectures about these objects. The author generalized
these objects and conjectures to the Fuss-Catalan level of generality. Even the
weakest of these conjectures would imply a collection of facts in
Coxeter-Catalan theory which are at present understood only in a case-by-case
fashion. We prove that when $W$ belongs to any the infinite families ABCDI, the
strongest of these conjectures is generically true.
