Evidence for parking conjectures

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.