  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.
