Is there a maximal finite depth infinite index irreducible subfactor?

Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley Lieb) ?

• A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$. Is there an infinite depth irreducible finite index maximal subfactor (other than Temperley-Lieb $A_{\infty} $) ?

• Answer:

  The infinite depth subfactor coming from SU(3) at any index above 9 gives an example. Here the Q-system is $V_{(1,0)} \otimes V_{(0,1)} \cong V_{(1,1)} \oplus V_{(0,0)}$ so the only possible sub-objects are the whole thing or the trivial, so it's certainly maximal.