Counterexamples to a rank analog of the Shepherd–Leedham-Green–Mckay theorem on finite $p$-groups of maximal nilpotency class

By the Shepherd–Leedham-Green–McKay theorem on finite $p$-groups of maximal nilpotency class, if a finite $p$-group of order $p^n$ has nilpotency class $n-1$, then $f$ has a subgroup of nilpotency class at most 2 with index bounded in terms of $p$. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in The Kourovka Notebook. Moreover, it is shown that there are no functions $r(p)$ and $l(p)$ such that any finite $2$-generator $p$-group whose all factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most $l(p)$ with quotient of rank at most $r(p)$. The required examples of finite $p$-groups are constructed as quotients of torsion-free nilpotent groups which are abstract $2$-generator subgroups of torsion-free divisible nilpotent groups that are in the Mal'cev correspondence with “truncated” Witt algebras.