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Conjecture: HD1 lies in BPP \ UPP

Prove or refute the conjecture that the 1‑Hamming Distance communication problem HD1 belongs to BPP but not to UPP (i.e., HD1 ∈ BPP \ UPP).

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Background

Resolving the membership of HD1 in BPP versus UPP would sharpen the separation between bounded‑ and unbounded‑error randomized communication classes and impact the characterization of UPP ∩ BPP. The conjecture originates from prior work and is tied to whether Equality reductions characterize UPP ∩ BPP.

References

This would imply UPP ⊄ BPP, and in particular the conjecture of [HHPTZ22] that HD_1 ∈ BPP\UPP.

— 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.”