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Integer Inner Product not in BPP (conjecture)

Prove that the Integer Inner Product functions, which belong to UPP and form a hierarchy in BPP under certain considerations, do not belong to BPP.

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Background

Integer Inner Product functions are known to lie in UPP and have been studied for hierarchies within BPP-related settings. The conjecture posits that these functions are not in BPP, which would influence our understanding of intersections between UPP and BPP and the limitations of constant-cost protocols.

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: Constant-Cost Communication: the Story so Far and Farther (Sign-rank)