Bi-orderability conjecture for fundamental groups of 3-manifolds

Let $M$ be a $3$-manifold and let $G=\pi_1(M)$ be its fundamental group. A nontrivial element of $G$ is a generalized torsion element if some nonempty finite product of its conjugates is the identity. The group $G$ is bi-orderable if it admits a total ordering invariant under multiplication from both the left and the right.

Bi-orderability 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 all $3$-manifolds. The converse is known for fundamental groups of non-hyperbolic geometric $3$-manifolds, but the general case remains open.