The Bishop–Phelps–Bollobás property and lush spaces

We prove that for every lush space X, the couple (ℓ1,X) has the Bishop–Phelps–Bollobás property for operators, that is, every lush space has the AHSP (standing for the approximate hyperplane series property). While every lush space has the alternative Daugavet property, there exists a space with the alternative Daugavet property that does not have the AHSP. We also show that there is a Banach space with both the AHSP and the alternative Daugavet property which is not lush.