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


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

