Relatively free, nearly nilpotent groups

We show that a free nilpotent group of countable rank, as well as a free group of countable rank of the variety defined by the identity $[[x_1,x_2,\dots,x_n],[x_{n+1},x_{n+2}]]=1$, satisfies the maximal condition for normal subgroups admitting endomorphisms induced by order preserving one-to-one mappings of the set of free generators into itself.