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?