Transverse knots with knot types of strongly quasi-positive knots

Question

In 2008, Etnyre and Van Horn-Morris proved that if $L$ is a fibered strongly quasi-positive link, there is a unique (up to transverse isotopy) transverse link with the knot type of $L$ in the standard tight contact structure of $S^3$ with maximal self-linking number.

What is known for (not necessarily fibered) strongly quasi-positive links? If $L$ is strongly quasi-positive, is there a classification of transverse links with the knot type of $L$, with maximal self-linking number? Any comments or references would be appreciated.

Answer 1

The twist knot $K_{-6} = m(7_2)$ is strongly quasipositive, not fibered, genus-1, and it's Legendrian non-simple; it has five Legendrian isotopy classes with Thurston-Bennequin number 1, rotation number 0. A good source of material for low-crossing knots is the Legendrian knot atlas, by Chongchitmate and Ng.

Legendrian twist knots have been classified by Etnyre, Ng, and Vértesi (J. Eur. Math. Soc. 15 (2013)). I doubt that there exists a classification for arbitrary strongly quasipositive knots.