The cup subalgebra has the absorbing amenability property

Consider an inclusion of diffuse von Neumann algebras A inside M. We say that the inclusion has the absorbing amenability property (AAP) if for any diffuse subalgebra B inside A and any amenable intermediate algebra D between B and M, we have that D is contained in A. We prove that the cup subalgebra associated to any subfactor planar algebra has the AAP.