A pronilpotent variety whose locally nilpotent subvarieties are nilpotent

Let \\(\\mathcal C\\) be the class of finite nilpotent metabelian groups whose Sylow \\(p\\)-subgroups have nilpotency class less than \\(p\\), for every prime \\(p\\). We prove that \\(\\operatorname{Pro}(\\mathcal C)\\) is a nonnilpotent variety of pronilpotent groups and that each of its locally nilpotent subvarieties is nilpotent. This gives a negative answer to Kourovka Problem 7.38(a). The proof combines a three-generator class-detection lemma for metabelian \\(p\\)-groups of class less than \\(p\\) with the finite groups \\[\\bigl(\\mathbb F_p[t]/(t^n)\\bigr)^+\\rtimes\\langle1+t\\rangle,\\qquad 1\\le n