Nonlinear maps preserving the second mixed triple η-*-product between von Neumann algebras

Let ? ? ?1 be a non-zero scalar, and let ? be a not necessarily linear bijection between two von Neumann algebras, one of which has no center abelian projections, satisfying ?(I) = I and ?(iI)* =-?(iI) and preserving the second mixed triple ?-*-product. It is showed that ? is a linear *-isomorphism if |?| = 1 and ? is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if |?|?1.