Conjecture on non-thickenable tori and supported contact structures

Let $K$ be a fibered knot in a manifold $M$, and let $\xi$ be a contact structure on $M$. Non-thickenable-tori conjecture. The knot type $K$ admits non-thickenable tori in $\xi$ if and only if $K$ supports $\xi$.

This conjecture seeks to characterize non-thickenable tori for fibered knots by the contact structures supported by their open books. The source gives no resolution status.