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 [3] and made three increasingly strong conjectures about these objects. The author generalized these parking objects and conjectures to the Fuss–Catalan level of generality [26]. Even the weakest of these conjectures would uniformly 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 of the infinite families ABCDI, the strongest of these conjectures is generically true.