We know that the set of Right-Orderable groups $RO$, is contained in the set of $\Omega$- groups (Read it from "A Note on Group Rings of Certain Torsion-Free Groups" Burns-Hale).

A Group $G$ is a right orderable group if there exists a total right order $\leq$ on $G$ s.t. if $ a \leq b$ $\implies$ $ga \leq gb$ $\forall g \in G$

A Group is an $\Omega$-group if for every ordered pair of nonempty finite subsets $A$, $B$ of $G$ there is at least one pair $(a,b)$ $\in$ $A \times B$ s.t. $ab\neq a'b'$ for any other ordered pair $(a',b') \in A \times B$.

So: Are all locally indicable groups $\Omega$-groups? and is there any proof relating the containment of locally indicable groups in $RO$-groups? (either up to some point or showing that they aren't contained?) Also what about containment of $\Omega$-groups in $RO$-groups?

Thanks : )

We know that the class of right-orderable groups $\mathit{RO}$, is contained in the class of $\Omega$-groups (read it from "A note on group rings of certain torsion-free groups" by Burns-Hale).

A group $G$ is a right-orderable group if there exists a total order $\leq$ on $G$ s.t. if $ a \leq b$ $\implies$ $ag \leq bg$ $\forall g \in G$

A group $G$ is an $\Omega$-group if for every ordered pair of nonempty finite subsets $A$, $B$ of $G$ there is at least one pair $(a,b)$ $\in$ $A \times B$ s.t. $ab\neq a'b'$ for any other ordered pair $(a',b') \in A \times B$.

A group is locally indicable if every nontrivial finitely generated subgroup admits $\mathbb{Z}$ as quotient group.

Are all locally indicable groups $\Omega$-groups? and is there any proof relating the containment of the class of locally indicable groups in the class of RO-groups? (either up to some point or showing that they aren't contained?) Also what about containment of the class of $\Omega$-groups in $\mathit{RO}$?