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.