Does k-Hamming Distance capture all of BPP up to reductions?

Determine whether the class of constant-cost public-coin randomized communication problems (BPP) is captured, up to constant-cost reductions, by Exact k-Hamming Distance for some fixed k; specifically, decide whether for every problem P ∈ BPP there exists a constant k such that D^{EHD_k}(P) = O(1).

Background

Although the paper proves that no single problem is complete for BPP, it remains open whether the family of k-Hamming Distance problems suffices to simulate all of BPP under constant-cost oracle reductions. The authors note they do not believe this is the case, but no counterexample is currently known.