  In this paper we study quasi-symmetric conjugations of $C^2$ weakly
order-preserving circle maps with a flat interval. Under the assumption that
the maps have the same rotation number of bounded type and that bounded
geometry holds we construct a quasi-symmetric conjugation between their
non-wandering sets. Further, this conjugation is extended to a quasi-symmetric
circle homeomorphism. Our proof techniques hinge on real-dynamic methods
allowing us to construct the conjugation under general and natural assumptions.
