On the relative commutants of subfactors

Let $A\subset B$ be factors generated by a periodic tower $A_{n}\subset B_{n}$ of finite dimensional $C^{*}$-algebras. We prove that for sufficiently large $n$, $A^{\prime }\cap B$ is $*$-isomorphic to a subalgebra of $A^{\prime }_{n}\cap B_{n}$.