Maps preserving semi-Fredholm operators on Hilbert C∗-modules

Let M be a Hilbert C∗-module over the C∗-algebra A, and LA(M) the C∗-algebra of all adjointable operators on M. Let φ:LA(M)→LA(M) be a unital linear map preserving semi-A-Fredholm operators in both directions and assume that φ is surjective up to compact operators. We will show that φ preserves compact operators and in certain cases, the induced map is an automorphism or an anti-automorphism.