Local indicability of left ordered groups without nonabelian free subgroups

Let $G$ be a group in the class $\operatorname{NF}$ of groups containing no nonabelian free subgroup. Recall that $G$ is left ordered if it admits a total ordering $\leq$ such that $x\leq y$ implies $gx\leq gy$ for all $g,x,y\in G$, and that it is locally indicable if every nontrivial finitely generated subgroup has an infinite cyclic quotient.

Local indicability conjecture. Every left ordered $\operatorname{NF}$-group is locally indicable.

This is presented as a stronger form of the question of whether every left ordered amenable group is locally indicable. The supplied text does not state whether the claim is resolved.