Let $N\subset M$ be a finite index inclusion of $II_1$ factors. To the inclusion we associate the tower of higher relative commutants

$\begin{array}{ccccccc} \mathbb{C} = N'\cap N & \subset & N'\cap M & \subset & N'\cap M_1 & \subset & \cdots \\ & & \cup & & \cup & & \\ & &\mathbb{C} = M'\cap M & \subset & M'\cap M_1 & \subset & \cdots \end{array} $

Where $M_1$ denotes the basic construction associated to $N\subset M$. We say that this inclusion is $\textit{finite depth}$ if the Bratteli diagram associated the the inclusion of the higher relative commutants has finite width (ie. The number of simple factors in the commutants is bounded).

My question is:

Is it possible to determine if a subfactor is finite depth by the growth rate of the dimension of $N'\cap M_k$? I know that this is bounded above by the index. If it attains this bound does that imply finite depth?