The generalized-torsion conjecture for 3-manifold groups

Let $G$ be the fundamental group of a $3$-manifold. A generalized torsion element is a non-trivial element for which some non-empty finite product of conjugates equals the identity. The group $G$ is bi-orderable if it admits a strict total ordering invariant under multiplication on both the left and the right.

Generalized-torsion conjecture. $G$ is bi-orderable if and only if $G$ has no generalized torsion element.

A bi-orderable group has no generalized torsion element, so the conjecture asserts the converse for fundamental groups of $3$-manifolds. The paper verifies it for non-hyperbolic geometric $3$-manifolds and for some infinite families of closed hyperbolic $3$-manifolds; it remains open in general.