An order automorphism of a Dlab group not induced by conjugation

Abstract

Let $G=D_{\langle2\rangle}([0,1])$ be equipped with either of its two Dlab orders. There exists an order automorphism $α$ of $G$ such that, for every rank-one subgroup $H\leq\mathbb R_{>0}^{\times}$, every one of the six corresponding Dlab groups $A$ listed below, equipped with any of its linear orders, every order-preserving embedding $e:G\hookrightarrow A$, and every $u\in A$, there exists $f\in G$ such that $e(α(f))\neq u^{-1}e(f)u$.