General Cambrian-like polytope and lattice construction for arbors

Construct, for every arbor t, a simple polytope Φ_t, an orientation of its skeleton whose Hasse diagram is a congruence-uniform Cambrian-like lattice Λ_t, and a noncrossing-like lattice NC_t obtained from Λ_t by the core-label-order construction, such that NC_t is a transmuted partner of P_t.

Background

The paper identifies a broad, unresolved extension of the relationship among generalized associahedra, Cambrian lattices, noncrossing partitions, and arbor posets. The desired construction should produce a dimension-n simple polytope with as many vertices as P_t has elements when t has size n. The authors explicitly state that no general method is currently known.