Displaceability conjecture for pre-Lagrangian toric fibers

Let $V$ be a compact toric contact manifold. A contact manifold is orderable if its identity component of the contactomorphism group admits no positive contractible loop, and a pre-Lagrangian toric fiber is a toric fiber that is pre-Lagrangian. Displaceability conjecture. If $V$ is not orderable, then all its pre-Lagrangian toric fibers are displaceable. This conjecture connects orderability with non-displaceability: known results show that suitable non-displaceable pre-Lagrangian fibers imply orderability, while the converse asserted here remains open.