On finite index infinite depth subfactors and reduction to depth 2

Question

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]?