BPP is not contained in P^NP (communication setting)

Show that the class BPP of polylogarithmic-cost public-coin randomized communication problems is not contained in P^NP, the class solvable by polylogarithmic-cost deterministic communication with NP-oracle queries charged by nondeterministic complexity.

Background

Impagliazzo–Williams proved strong lower bounds for P^NP via weighted rectangle ratio, and known BPP examples (e.g., Integer Inner Product) suggest BPP may transcend P^NP. However, no unconditional separation is known in the communication-class setting.

A separation would mark a major advance in the structural understanding of randomized vs. oracle-aided deterministic communication.