Do k-Hamming Distance oracles capture all of BPP (constant cost)?

Determine whether every communication problem with public-coin constant-cost randomized complexity reduces, via a constant-cost deterministic oracle protocol, to Exact k-Hamming Distance (EHD_k) for some constant k. Equivalently, decide whether for every problem P ∈ BPP there exist constants k and c and a Boolean function f such that for all N there are Q_1,…,Q_c in QS(EHD_k) with P_N(x,y) = f(Q_1(x,y),…,Q_c(x,y)).

Background

The paper develops new Ramsey-theoretic techniques to prove separations among BPP problems and establishes that no single problem is complete for BPP. It also proves an infinite hierarchy within k-Hamming Distance.

Despite these advances, the authors note that it remains open whether the union of all k-Hamming Distance problems, as oracles under constant-cost reductions, suffices to express every problem in BPP.