This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.)

The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of Young diagrams. (The sizes of the diagrams are arbitrary and need not be the same. The diagram $\lambda_+$ corresponds to upward oriented strands, while $\lambda_-$ corresponds to downward oriented strands.) Consequently one can define numerical invariants of oriented links whose components are labeled by pairs of Young diagrams. This is the "colored" HOMFLY-PT polynomial.

Of fundamental importance in this subject are the invariants $S_{\lambda_+\lambda_-,\mu_+\mu_-}$ of the Hopf link with its components labeled by pairs of Young diagrams (i.e. idempotents) $(\lambda_+, \lambda_-)$ and $(\mu_+, \mu_-)$. In TQFT language, this is the "S-matrix" of the theory.

My Question:

Has the S-matrix for the HOMFLY-PT category been calculated and published? If not, are partial results in this direction known?

I am aware of this paper by Morton and Lukac, which does the case where $\lambda_-$ and $\mu_-$ are both empty (i.e. all strands oriented the same direction). This paper by Morton and Hadji is also related. Are there other relevant papers that I have missed?

See also the BMW version of this question here.