Conjecture: Bounded sign-rank is preserved under constant-cost reductions

Prove that if communication problems P and Q satisfy D^Q(P)=O(1) (a constant-cost deterministic oracle reduction from Q to P) and Q has bounded sign-rank, then P also has bounded sign-rank.

Background

Sign-rank is a fundamental matrix-analytic complexity measure with tight connections to communication complexity. Understanding how sign-rank behaves under reductions is important for transferring lower bounds and structural properties across problems.

The conjecture would, in particular, relate the structure of problems reducible to those with bounded sign-rank (e.g., IIP_d is known to have bounded sign-rank) and would yield consequences for separating classes and for lower bounds.