Let $(N \subset M)$ be a finite depth-index irreducible subfactor.
Main question: Is $(N \subset M)$ the intermediate of a finite index depth $\le 3$ irreducible subfactor?
(In others words, is there $P \subset N$ or $Q \supset M$ such that $(P \subset M)$ or $(N \subset Q)$ is a finite index depth $\le 3$ irreducible subfactor?)
Remark: By the prop. 9.1.1 p 37 here, an irreducible finite depth-index subfactor is always an intermediate subfactor of a depth $2$ reducible subfactor. But, this post asks if it is always the intermediate of a finite index depth $\le 3$ irreducible subfactor.
Remark: the first examples to look at should be the $A_n$-subfactors (of depth $n-1$).
It's ok for $n=2,3,4$; for $n=5$ also: it's isomorphic to $(R^{S_3} \subset R^{S_2})$, intermediate of $(R^{S_3} \subset R)$.
Question: Is $A_6$ a counter-example? If yes, why? Else, is it true for all $n \ge 2$ ?
Definition: A finite index depth $d$ irreducible subfactor $(N \subset M)$ is called top if it can't be the intermediate of a strictly smaller depth subfactor (in others words, such a subfactor is top if $\forall P \subset N$ and $\forall Q \supset M$ such that $(P \subset M)$ and $(N \subset Q)$ are irreducible, then their depths are $\ge d$).
Examples: A group-subgroup subfactor is top iff it's a group subfactor.
The non-integer index depth $3$ irreducible subfactors are top.
The question Are the integer index finite depth irreducible subfactors Kac-coideal? is reformulable by: Are the integer index finite depth irreducible top subfactors, depth $2$?
The main question is also reformulable by: Is a finite depth-index irreducible top subfactor, depth $\le 3$?
Optional part: top-simple reduction
If the main question admits a negative answer:
Question: Is there an irreducible finite index top subfactor of arbitrary large depth $d$?
If yes, is there one for all $d \ge 2$?
Remark: The classification of the subfactors reduces to the classification of the top subfactors, because all the others come as intermediates.
Let $(N \subset M)$ be a finite depth-index irreducible top subfactor, and $(N \subset P \subset M)$ an intermediate.
Question: $P$ normal $\Rightarrow$ depth$(N \subset P)$, depth$(P \subset M) \le$ depth$(N \subset M)$?
Remark: For the depth $2$ case, it's true and the converse also. Is it also an equivalence at depth $d$?
Question: Does the Jordan-Hölder theorem work for the finite depth-index irreducible top subfactors?
Remark: this anwser shows that, without the top assumption, Jordan-Hölder theorem does not work, counter-examples are given the $(A_n \subset S_{n+1})$ group-subgroup subfactors (for $n \ge 3$).
If the two previous questions admit positive answers then the classification of the depth $\le d$ finite index irreducible subfactors reduces to the top simple ones.
