  In this note we analyze displaceability of pre-Lagrangian toric fibers in
contact toric manifolds. While every symplectic toric manifold contains at
least one non-displaceable Lagrangian toric fiber and infinitely many
displaceable ones, we show that this is not the case for contact toric
manifolds. More precisely, we prove that for the contact toric manifolds
$\mathbb{S}^{2d-1} (d\geq 2)$ and $\mathbb{T}^k \times \mathbb{S}^{2d+k-1} (d
\geq 1)$ all pre-Lagrangian toric fibers are displaceable, and that for all
contact toric manifolds for which the toric action is free, except possibly
non-trivial $\mathbb{T}^3$-bundles over $\mathbb{S}^2$, all pre-Lagrangian
toric fibers are non-displaceable. Moreover we also prove that if for a compact
connected contact toric manifold all but finitely many pre-Lagrangian toric
fibers are non-displaceable then the action is necessarily free. On the other
hand, as we will discuss, displaceability of all pre-Lagrangian toric fibers
seems to be related to the non-orderability of the underlying contact
manifolds.
