This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)

The minimal idempotents of this category are indexed by Young diagrams (of arbitrary size; there are infinitely many of them). Consequently one can define numerical invariants of unoriented links whose components are labeled by Young diagrams. This is the "colored" BMW/2K polynomial.

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

My Question:

Has the S-matrix for the BMW / 2-variable Kauffman category been calculated and published? If not, are partial results in this direction known?

I've done some searching, but so far I've not found anything.

See also the HOMFLY-PT version of this question here.