Binding number conjecture for Giroux torsion

Let $(Y,\xi)$ be a contact 3-manifold. The binding number $bn(\xi)$ is the minimal number of binding components among open books supporting $\xi$ whose pages have minimal genus. The Giroux torsion of $\xi$ is the largest $n$ for which there is a contact embedding

$$(T^2 \times I,\xi_{2n\pi})\longrightarrow (Y,\xi),$$

where $\xi_{2n\pi}=\ker(\cos(2n\pi t)\,dx+\sin(2n\pi t)\,dy)$. Binding number conjecture. The binding number $bn(\xi)$ of a tight contact structure is bounded below by the Giroux torsion of $\xi$. This would relate a combinatorial invariant of supporting open books to Giroux torsion, which is currently not visible from the open-book perspective; the source gives no resolution.