The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right-orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then $ac<bc$ for all $c\in G$. It is known that braid groups are right-orderable.

Is there a non-right-orderable torsion-free quotient group of the braid group on 3 strands?