Nonlinear maps preserving asymptotic equivalence on B(X)

Let X be an infinite-dimensional complex Banach space. By B(X) we denote the algebra of all bounded linear operators on X. It is shown that if Phi : B(X) -> B(X) is a surjective map satisfying A + B is asymptotically equivalent to C if and only if Phi(A) + Phi(B) is asymptotically equivalent to Phi(C), then either there exist bijective continuous linear or conjugate-linear maps T, S : X -> X such that Phi(A) = TAS for every A is an element of B(X), or there exist bijective continuous linear or conjugate-linear maps T : X* -> X and S : X -> X * such that Phi(A) = TA* S for every A is an element of B(X).