Let $(N \subseteq M)$ be a finite index unital inclusion of ${\rm II}_1$ factors. Let $e_1$ be the Jones' projection.
A finite subset $\{\lambda_i, i \in I\} \subset M $ is called a (right) Pimsner-Popa basis if $ \sum_{i \in I} \lambda_i^* e_1 \lambda_i = 1$.
Question: Is there a Pimsner-Popa basis $\{\mu_j, j \in J\}$ for $(M' \subseteq N')$ such that the following holds?
$$\forall j \in J, \exists k \ge 0 \text{ such that } \mu_j \in N' \cap M_k$$
Bonus question: If so, can we restrict to $k \le d=\text{depth}(N \subseteq M)$?
In other words, is there a Pimsner-Popa basis $\mathcal{B}$ for $(M' \subseteq N')$ such that $\mathcal{B} \subset M_d$?
