Conjecture: Unbounded sign-rank of Exact 1-Hamming Distance

Prove that the Exact 1-Hamming Distance problem EHD_1 has unbounded sign-rank; that is, the sign-rank of the matrices EHD_1^n grows without bound as n increases.

Background

Exact 1-Hamming Distance (EHD_1) is a central problem in the paper’s paper of constant-cost randomized communication. Establishing unbounded sign-rank for EHD_1 would have strong implications for lower bounds and would interact with the conjectured preservation of bounded sign-rank under constant-cost reductions.

This conjecture appears in prior work and, together with the previous conjecture, would imply stronger separations than currently known.