Radicals and Plotkin's problem concerning geometrically equivalent groups

If G and X are groups and N is a normal subgroup of X, then the G-closure of N in X is the normal subgroup (X) over bar (G) = boolean AND {ker phi \ phi : X --> G; with N subset of or equal to ker phi} of X. In particular, (1) over bar (G) = RGX is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G similar to H, if for any free group F of finite rank and any normal subgroup N of F the G-closure and the H-closure of N in F are the same. Quasi-identities are formulas of the form (/\(i less than or equal ton) w(i) = 1 --> 2 = 1) for any words w, w(i) (i less than or equal to n) in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups G and H satisfy the same quasi-identities if and only if G and H are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.