Bounded Generation of Submonoids of Heisenberg Groups

If G is a nilpotent group and $[G,G]$ has Hirsch length $1$, then every f.g. submonoid of G is boundedly generated, i.e. a product of cyclic submonoids. Using a reduction of Bodart, this implies the decidability of the submonoid membership problem for nilpotent groups G where $[G,G]$ has Hirsch length $2$.