Torsion vanishing conjecture for the Ozsváth–Szabó contact invariant

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. Let ${\rm Tor}(Y,\xi)$ denote the torsion, defined as the supremum of the integers $n\geq 1$ for which there exists a contact embedding

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

with ${\rm Tor}(Y,\xi)=0$ if no such embedding exists. Let $c(\xi)$ denote the Ozsváth–Szabó contact invariant. Torsion vanishing conjecture. If

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

then $c(\xi)=0$. The source presents this as a stronger plausible statement than the obstruction to strong symplectic fillability; its general validity remains open.