Jordan product and analytic core preservers

Let $\mathcal{B}(X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. For an operator $T\in \mathcal{B}(X)$, $K(T)$ denotes as usual the analytic core of $T$. We determine the form of surjective maps $\phi$ on $\mathcal{B}(X)$ satisfying $$K(\phi (T)\phi (S)+\phi (S)\phi (T))=K(TS+ST)$$ for all $T,S\in \mathcal{B}(X)$.