Eliashberg's torsion obstruction to strong symplectic fillability

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. The torsion of $(Y,\xi)$, denoted by

$${\rm Tor}(Y,\xi),$$

is the supremum of the integers $n\geq 1$ for which there exists a contact embedding

$$\bigl(T^2\times[0,1],\,\ker(\cos(2n\pi z)dx-\sin(2n\pi z)dy)\bigr)\hookrightarrow(Y,\xi).$$

If no such embedding exists, set ${\rm Tor}(Y,\xi)=0$. Eliashberg's torsion conjecture. If

$${\rm Tor}(Y,\xi)>0,$$

then $(Y,\xi)$ is not strongly symplectically fillable. This conjecture is motivated by the examples in the paper, where positive torsion obstructs strong symplectic fillability; the general implication remains open.