Unbounded sign-rank of Exact 1-Hamming Distance (EHD_1)

Prove that the Exact 1-Hamming Distance communication problem EHD_1 has unbounded sign-rank.

Background

The sign-rank of EHD_1 is a long-standing structural question. Establishing that EHD_1 has unbounded sign-rank would, together with preservation under constant-cost reductions (if true), yield stronger separations between EHD_1 and problems known to have bounded sign-rank, such as IIP_d for fixed d.