A commutator description of the solvable radical of a finite group

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of g ∈ G such that for any k elements a1, a2, . . . , ak ∈ G the subgroup generated by the elements g, aiga−1 i , i = 1, . . ., k, is solvable. We consider a similar problem of finding the smallest integer ℓ > 1 with the property that R(G) coincides with the collection of g ∈ G such that for any ℓ elements b1, b2, . . . , bℓ ∈ G the subgroup generated by the commutators [g, bi ], i = 1, . . . , ℓ, is solvable. Conjecturally, k = ℓ = 3. We prove that both k and ℓ are at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup