Bound sign-rank under conjunction of matrices

Determine whether the sign-rank of the entrywise conjunction A ∧ B can be bounded in terms of the sign-ranks of A and B; i.e., prove or refute the existence of a general bound signrank(A ∧ B) ≤ f(signrank(A), signrank(B)).

Background

The authors emphasize that standard boolean combinations pose challenges for sign-rank analysis. Even if one can bound sign-rank for components (e.g., HD≤k), existing techniques do not transfer to conjunctions due to the lack of known closure or bounding principles.

A resolution would significantly advance structural understanding of sign-rank and enable modular analyses of complex matrices built from simpler ones.