Let $(N \subset M)$ be an inclusion of ${\rm II}_1$ factors, the basic construction is $N \subset M \subset M_1 = \langle M , e^M_N \rangle$.

Question: For any intermediate subfactor $N \subset P \subset M_1$, is it true that $N \subset P \subseteq M$ or $M \subset P \subset M_1$ (i.e. no extra intermediate)?


This answer came by a discussion with Keshab Chandra Bakshi.
Consider the subfactor $(R \subset R \rtimes \mathbb{Z}/2)$, then $R \subset R \rtimes \mathbb{Z}/2 \subset M_2(R) = R \otimes M_2(\mathbb{C})$ is the basic construcition, but $(R \subset M_2(R))$ admits continuously many intermediate subfactors which are conjugate to $R \rtimes \mathbb{Z}/2$ by unitaries in $\mathbb{C} \otimes M_2(\mathbb{C})$. So the answer is no.

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