Eventual uniqueness of standard neighborhoods for transverse knot representatives

Fix a smooth knot type $K$. A transverse representative $T$ of $K$ has a standard neighborhood, whose boundary has a slope, and standard neighborhoods are considered up to contact isotopy. Eventual neighborhood uniqueness conjecture. There is an integer $n$ such that, whenever $T$ is a transverse representative of $K$ with $\self(T)<n$, any standard neighborhood of $T$ is determined up to contact isotopy by its boundary slope. This conjecture proposes eventual uniqueness among sufficiently low-self-linking transverse representatives of each fixed knot type; its general validity remains open.