Bounded nilpotency class for finitely generated d-maximal 2-groups

Let $r\geq 1$ be an integer. A finite $2$-group $G$ is $d$-maximal if $d(H)<d(G)$ for every proper subgroup $H<G$, where $d(G)$ is the minimal number of generators. The bounded-class conjecture. For every integer $r\geq 1$, the nilpotency class of an $r$-generated $d$-maximal $2$-group is bounded in terms of $r$.

The conjecture is equivalent, using the lemma mentioned in the source, to the assertion that the orders of $r$-generated $d$-maximal $2$-groups are bounded in terms of $r$. It is motivated by the observation that known $d$-maximal $2$-groups of higher class require more generators; the supplied text gives no evidence that the conjecture has been resolved.