An endomorphism semigroup α is greater than an endomorphism semigroup β if β is cocycle conjugate to a subordinate of α. Using results of R.T. Powers, G.L. Price and the speaker on the computation of the gauge group, we will present a proof that, in the case of endomorphism semigroups arising from one-dimensional boundary weights, this relation is antisymmetric: if α is greater than β and β is greater than α then α and β are cocycle conjugate.