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Conjecture: Integer Inner Product functions are not in BPP

Prove or refute the conjecture that the Integer Inner Product communication functions do not belong to BPP.

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Background

Integer Inner Product functions are known to lie in UPP and play a central role in hierarchies studied in communication complexity. Showing they are not in BPP would reinforce separations between classes and constrain the search for Equality‑reducible characterizations of UPP ∩ BPP.

References

It would also imply the conjecture of [CHHS23] that the Integer Inner Product functions (which belong to UPP and form a hierarchy in BPP [CLV19]) do not belong to BPP.

— Constant-Cost Communication is not Reducible to k-Hamming Distance (2407.20204 - Fang et al., 29 Jul 2024) in Section 1.5 (Constant-Cost Communication: the Story so Far and Farther), paragraph “Sign-rank.”