Rank-boundedness conjecture for finite 2-groups with few rational characters

Let $G$ be a finite $2$-group, let $\operatorname{rk}(G)$ denote its rank, and let $q$ be the number of rational characters of $G$. Rank-boundedness conjecture. The rank $\operatorname{rk}(G)$ is $q$-bounded, meaning that it is bounded above by a function depending only on $q$. This is proposed as an analogue of a preceding result involving real characters; the source gives no resolution, so the boundedness assertion remains open.