Uniqueness of standard neighborhoods with negative boundary slope

Let $T$ be a transverse knot in a contact manifold. A standard neighborhood of $T$ has a boundary slope, measured with respect to the chosen framing, and standard neighborhoods are considered up to contact isotopy. Negative-slope neighborhood conjecture. Any standard neighborhood of $T$ with boundary slope less than zero is unique up to contact isotopy. This conjecture is motivated by the known examples of transverse knots with non-unique standard neighborhoods, whose non-uniqueness occurs at nonnegative slopes; the general uniqueness assertion remains open.