The Baer–Suzuki width of a complete class of finite group is finite

Let $\mathscr{X}$ be a nonempty class of finite groups closed under the taking subgroups, homomorphic images and extensions. According to Gordeev, Grunewald, Kunyavskiĭ, and Plotkin, the Baer-Suzuki width $\mathrm{BS}(\mathscr{X})$ of $\mathscr{X}$ does not exceed a non-negative integer $m$ if, in any finite group $G$, the largest normal $\mathscr{X}$-subgroup coincides with the set of elements $x$ such that every $m$ elements conjugate to $x$ generate an $\mathscr{X}$-subgroup. If there are no $m$ for which ${\mathrm{BS}(\mathscr{X})\leqslant m}$, then by definition ${\mathrm{BS}(\mathscr{X})=\infty}$. In the paper, it is proved that ${\mathrm{BS}(\mathscr{X})<\infty}$ for every class $\mathscr{X}$ with the above properties. More precisely, if $\mathscr{X}$ is distinct from the class of all finite groups, then the value of $\mathrm{BS}(\mathscr{X})$ does not exceed $\max\{11,2\Upsilon+1\}$ where $\Upsilon$ is equal to the largest $n$ such that ${\mathrm{Sym}_n\in\mathscr{X}}$.