Mappings preserving sum of products $a\circ b+ba^{*}$ on factor von Neumann algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras. In this paper, we proved that a bijective mapping $\Phi \colon\mathcal{A}\to\mathcal{B}$ satisfies $\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}$ (where $\circ $ is the special Jordan product on $\mathcal{A}$ and $\mathcal{B},$ respectively), for all elements $a,b\in \mathcal{A}$, if and only if $\Phi $ is a $\ast $-ring isomorphism. In particular, if the von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$ are type I factors, then $\Phi $ is a unitary isomorphism or a conjugate unitary isomorphism.