A commutator description of the solvable radical of a finite group

We are looking for the smallest integer k>1 providing the followingcharacterization of the solvable radical R(G) of any finite group G: R(G)coincides with the collection of all g such that for any k elementsa_1,a_2,...,a_k the subgroup generated by the elements g, a_iga_i^{-1},i=1,...,k, is solvable. We consider a similar problem of finding the smallestinteger l>1 with the property that R(G) coincides with the collection of all gsuch that for any l elements b_1,b_2,...,b_l the subgroup generated by thecommutators [g,b_i], i=1,...,l, is solvable. Conjecturally, k=l=3. We provethat both k and l are at most 7. In particular, this means that a finite groupG is solvable if and only if in each conjugacy class of G every 8 elementsgenerate a solvable subgroup.