Relative bicentralizer conjecture for maximal abelian subalgebras with expectation

Let $N \subset M$ be an irreducible inclusion of factors with expectation and with separable preduals, meaning that $N' \cap M=\mathbb{C}$ and there is a faithful normal conditional expectation from $M$ onto $N$. Let $c(N)$ and $c(M)$ denote the continuous cores, and let $A \subset N$ be a maximal abelian subalgebra with expectation when there is a faithful normal conditional expectation from $N$ onto $A$. Relative bicentralizer conjecture. The following are equivalent: (1) there exists a maximal abelian subalgebra with expectation $A \subset N$ that is also maximal abelian in $M$; (2) $N' \cap c(M) \subset c(N)$. This gives an if-and-only-if criterion for the existence of the desired maximal abelian subalgebra; the paper proves it for a large class of inclusions, but the conjecture is not stated as resolved in full.