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 X̄^G = ∩ {ker φ|φ X → G, with N ⊂ ker φ} of X. In particular, 1̄^G = R_GX is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G ∼ 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 (Λ_i≤n w_i = 1 → w = 1) for any words w, w_i (i ≤ 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.