Robinson's solvability conjecture for finite groups with few Sylow subgroups

Let $G$ be a finite group, and let $v_p(G)$ denote the number of Sylow $p$-subgroups of $G$. Robinson's solvability conjecture. If

$$v_p(G)\leq p^2-p+1$$

for each odd prime number $p$, then $G$ is solvable. The paper gives an affirmative answer to this conjecture by proving the stronger result that a finite group containing at most $p^2$ Sylow $p$-subgroups for each odd prime $p$, and at most four Sylow $2$-subgroups, is solvable.