The first higher Stasheff-Tamari orders are quotients of the higher Bruhat orders

We prove the conjecture that the higher Tamari orders of Dimakis and M"uller-Hoissen coincide with the first higher Stasheff–Tamari orders. To this end, we show that the higher Tamari orders may be conceived as the image of an order-preserving map from the higher Bruhat orders to the first higher Stasheff–Tamari orders. This map is defined by taking the first cross-section of a cubillage of a cyclic zonotope. We provide a new proof that this map is surjective and show further that the map is full, which entails the aforementioned conjecture. We explain how order-preserving maps which are surjective and full correspond to quotients of posets. Our results connect the first higher Stasheff–Tamari orders with the literature on the role of the higher Tamari orders in integrable systems.