Real ℓ_∞ satisfies BPBp-nu while complex ℓ_1 ⊕_∞ c_0 does not, supplying the first natural example where numerical-radius-attaining operators are dense yet BPBp-nu fails.
Bourgain,On dentability and the Bishop-Phelps property, Israel J
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The Bishop-Phelps-Bollob\'as property for the numerical radius: a Zizler-type approach
Real ℓ_∞ satisfies BPBp-nu while complex ℓ_1 ⊕_∞ c_0 does not, supplying the first natural example where numerical-radius-attaining operators are dense yet BPBp-nu fails.