Non-constant cost conjecture for Integer Inner Product (IIP_d)

Prove that for each fixed constant dimension d, the Integer Inner Product communication problem IIP_d does not admit a constant-cost public-coin randomized protocol; equivalently, show that the randomized communication complexity R(IIP_d^n) grows unboundedly with n (i.e., IIP_d ∉ BPP as defined in the paper).

Background

The Integer Inner Product family IIP_d (for fixed d) is a canonical set of problems in randomized communication. It is known that R(IIP_d^n)=O(d·log n), so each IIP_d lies in the broader polylogarithmic class (the paper’s BPP), but whether they admit constant-cost protocols (the paper’s BPP) has remained unresolved.

The paper uses IIP_d as a benchmark to separate it from k-Hamming Distance under certain reductions, but it does not resolve the fundamental conjecture that IIP_d requires non-constant cost.