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 $T, S \in \mathcal{B} (X)$.