Reduced-graph order bound by rank

Prove that for every integer r≥2, every reduced graph of rank r has order at most m(r), where m(r)=2^{(r+2)/2}−2 for even r and m(r)=5·2^{(r−3)/2}−2 for odd r.

Background

The paper reviews a conjecture attributed to Akbari, Cameron, and Khosrovshahi concerning the maximum order of a reduced graph in terms of adjacency-matrix rank. The conjecture is supported by constructions attaining the proposed bound, while subsequent work limits a possible counterexample to rank at most 46.