The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was introduced by Edelman and Reiner, who also conjectured the two to coincide in 1996. In this paper we prove their conjecture, thereby substantially increasing our understanding of these orders. This result also has ramifications in the representation theory of algebras, as established in previous work of the author. Indeed, it means that the two corresponding orders on tilting modules, cluster-tilting objects and their maximal chains are equal for the higher Auslander algebras of type $A$.