On one-sided orders in groups with ascending central series

It is proved, that for the right-ordered $Z-A$-group $Q$ the following four properties are equivalent:
1 ) the group $Q$ is archimedean,
2 ) the group $Q$ has no proper convex subgroups,
3 ) in the group $Q$ all abelian subgroups are archimedean,
4) the group $Q$ has the archimedean embedded centre $Z$, i.e . $(\forall q\in Q, \forall z\in Z)\ q>z>1\to (\exists n>0)\ z^n>q$.
In the paper [1] it was demonstrated the example of the right-ordered metabelian group, which has the properties 2) and 3), but is not archimedean.