The reduced-graph order conjecture of Akbari, Cameron, and Khosrovshahi

Let $r\geq 2$, and let a reduced graph mean a graph with no pair of vertices that can be removed without changing the relevant adjacency-matrix rank. Write $m(r)$ for

$$m(r)=\begin{cases}2^{(r+2)/2}-2&\text{if }r\text{ is even},\\5\cdot2^{(r-3)/2}-2&\text{if }r\text{ is odd}. \end{cases}$$

Akbari–Cameron–Khosrovshahi conjecture. The order of any reduced graph of rank $r$ is at most $m(r)$. This conjecture is presented as a bound on the order of reduced graphs in terms of adjacency-matrix rank; the paper cites it as a proposal of Akbari, Cameron, and Khosrovshahi, without giving a resolution.