Let $(N \subset M)$ be a finite index irreducible subfactor (with $N$ and $M$, ${\rm II}_1$ factors).
Let $N \subset M \subset M_1 \subset M_2 \subset \cdots$ be the tower of basic constructions.

If $(N \subset M)$ is of depth $n$, then $(N \subset M_{n-2})$ is of depth $2$ (see Prop. 9.1.1 p37 here).

Now if $(N \subset M)$ is infinite depth. Let $M_{\infty} = (\bigcup M_i)'' \subset B(L^2(M,tr))$.
[Is $M_{\infty}$ a factor? which type?]

Question: Is $(N \subset M_{\infty})$ of depth $2$?
[in the sense that every irreducible $N$-$N$-bimodule coming from relative tensor products of $M_{\infty}$ as $N$-$M$ and $M$-$N$-bimodule, are sub-$N$-$N$-bimodule of $M_{\infty}$ as $N$-$N$-bimodule]

Let $(N \subset M)$ be a finite index irreducible subfactor (with $N$ and $M$, ${\rm II}_1$ factors).
Let $N \subset M \subset M_1 \subset M_2 \subset \cdots$ be the tower of basic constructions.

If $(N \subset M)$ is of depth $n$, then $(N \subset M_{n-2})$ is of depth $2$ (see Prop. 9.1.1 p37 here).

Now if $(N \subset M)$ is infinite depth. Let $M_{\infty} = (\bigcup M_i)'' \subset B(L^2(M,tr))$.
[Is $M_{\infty}$ a factor? which type?]

Question: Is $(N \subset M_{\infty})$ of depth $2$ [in the sense of bimodules]?