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.

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., 2024) in Section 1.5 (Constant-Cost Communication: the Story so Far and Farther), paragraph “Sign-rank.”