Three-conjugate characterization of the solvable radical

Let $G$ be a finite group, and let $R(G)$ denote its solvable radical, the largest solvable normal subgroup of $G$. For $g,a,b,c\in G$, consider the subgroup generated by $g$ and its conjugates by $a$, $b$, and $c$. Three-conjugate solvable-radical conjecture. The solvable radical of $G$ coincides with the collection of elements $g\in G$ such that, for every $a,b,c\in G$, the subgroup

$$\langle g,aga^{-1},bgb^{-1},cgc^{-1}\rangle$$

is solvable. This would sharpen the proved characterization using seven conjugates and would imply that a finite group is solvable if and only if every four conjugate elements generate a solvable subgroup; the source does not report a resolution.