Do reductions preserve sign-rank?

Determine whether sign-rank is preserved under communication reductions: specifically, prove or refute that if a boolean matrix family F has bounded support-rank (or sign-rank), then any boolean matrix computable by a constant-depth deterministic protocol with oracle queries to F also has bounded sign-rank.

Background

The paper’s technique gives constant sign-rank bounds for matrices obtained from k-Hamming Distance by reductions and compositions. However, they note that whether reductions preserve sign-rank is a well-known open problem, so their approach circumvents rather than resolves this general closure question.

Resolving this would clarify the robustness of sign-rank under standard oracle reductions and help systematize which classes of matrices inherit constant sign-rank from simpler primitives.