The word problem in the variety of inverse semigroups with Abelian covers

The variety of inverse semigroups which possess E-unitary covers over Abelian groups coincides with the Mal'cev product of the variety of semilattices and the variety of Abelian groups, and also with the variety generated by semidirect products of semilattices and Abelian groups. We show that this variety (and any variety of inverse semigroups that contains this variety) has undecidable word problem.