Some kind of Bishop–Phelps–Bollobás property

In this paper we introduce two Bishop–Phelps–Bollobás type properties for bounded linear operators between two Banach spaces X and Y : property 1 and property 2. These properties are motivated by a Kim–Lee result which states, under our notation, that a Banach space X is uniformly convex if and only if the pair satisfies property 2. Positive results of pairs of Banach spaces satisfying property 1 are given and concrete pairs of Banach spaces failing both properties are exhibited. A complete characterization of property 1 for the pairs is also provided.