TITLE: realizing fusion categories as subfactors of the hyperfinite
QUESTION [8 upvotes]: Let R be the hyperfinite II_1 or the hyperfinite III_1 factor (pick which ever one you prefer), and let Bim(R) denote the tensor category of R-R-bimodules.

This question is inspired by the recent article 
The classification of subfactors of index at most five
by Jones, Morrison, and Snyder.
More precisely, I am interested in Thm 1.1 and Thm 2.10 of the above paper (both of them are older results).
Given the above,
I expect the following to be true:

(1) Any unitary fusion category can be embedded in Bim(R).
  (2) Any two embeddings are conjugated by an automorphism of R.

however, I am not sure if things have ever been formulated in this way.
My questions are:
• What is the closest result to (1) and (2) available in the literature?
• is it easy to adapt/use some existing proofs to get (1) and (2), and how?

REPLY [6 votes]: 1, and indeed its generalization to the amenable case, is in "Amenable tensor categories and their realizations as AFD bimodules" by Hayashi and Yamagami, see Section 7.  I don't think 2 has appeared in the literature, though I'd expect that it's true.
In the fusion case, if the universal grading group is trivial, I think you can just look at the algebra object $\oplus_x x \otimes x^*$ in $\mathcal{C}$ and use the usual reconstruction and uniqueness for subfactors.  It's possible I'm missing a technicality, but I think this should work (in particular, Popa doesn't assume irreducibility).  If the grading group is non-trivial there still may be some trick that reduces it to the known cases, but it's not clear to me.