Teragaito–Motegi conjecture on bi-orderability and generalized torsion in 3-manifold groups

Let $G$ be the fundamental group of a $3$--manifold. A generalized torsion element is a non-trivial element whose non-empty finite product of conjugates is the identity, and $G$ is bi-orderable if it admits a total ordering invariant under left and right multiplication. Teragaito–Motegi conjecture. The group $G$ is bi-orderable if and only if $G$ has no generalized torsion element.

This conjecture connects bi-orderability of $3$-manifold groups with the existence of generalized torsion. It was verified for non-hyperbolic geometric $3$-manifolds and some other examples, but its general status is not resolved in the supplied text.