Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.

There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < S_5$, and others. There's exactly one known example of a $5$-ST subfactor, the Haagerup-Asaeda subfactor, and one $7$-ST subfactor, the extended Haagerup subfactor.

Below index $4$ there are the $A_n$ and $D_n$ families, which are arbitrarily super-transitive. Ignore those; I'm just interested above index $4$.

Is there anything that's even more super-transitive?