The Bishop–Phelps–Bollobás property for numerical radius on L1L1

Guirao and Kozhushkina recently introduced the Bishop–Phelps–Bollobás property for numerical radius. Geometrically speaking, the Bishop–Phelps–Bollobás property says that if we have a state and an operator that almost attains its numerical radius at this state, then there exist another state close to the original state and another operator close to the original operator, such that the new operator attains its numerical radius at this new state. In this paper we study the Bishop–Phelps–Bollobás property for numerical radius in the context of the Banach space of Lebesgue integrable functions over the real line.