Unification types of completely regular semigroups

We show that a variety V of completely regular semigroups that contains a nontrivial (two-element) semilattice has finitary or unitary unification type if and only if V consists of strong semilattices of rectangular groups of finitary or unitary unification type. We reduce the problem of recognizing unification type of a variety V of completely regular semigroups defined by a homotypical system of identities to the same problem for the variety of all groups of V.