Nonlinear maps preserving the mixed Jordan triple $eta$-$*$-product between factors

Let $mathcal{A}$ and $mathcal{B}$ be two factor von Neumann algebras and $eta$ be a non-zero complex number. A nonlinear bijective map $phi:mathcal Arightarrowmathcal B$ has been demonstrated to satisfy $$phi([A,B]_{*}^{eta}diamond_{eta} C)=[phi(A),phi(B)]_{*}^{eta}diamond_{eta}phi(C)$$ for all $A,B,Cinmathcal A.$ If $eta=1,$ then $phi$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $eta eq 1$ and satisfies $phi(I)=1,$ then $phi$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.