Regular unimodular triangulation conjecture for the parking function polytope

Let $\mathsf{PF}_n$ denote the parking function polytope. A triangulation of a lattice polytope is regular if it is induced by a lifting function, and unimodular if every simplex in it has normalized volume one. Regular unimodular triangulation conjecture. The parking function polytope $\mathsf{PF}_n$ admits a regular unimodular triangulation. This conjecture suggests an approach via Gröbner bases: by the correspondence between regular triangulations and initial ideals of toric ideals, one may seek a term order whose initial ideal is square-free, which is equivalent to the associated regular triangulation being unimodular.