Bardakov–Iskra conjecture on horizontal class transposition groups

Let $n>1$. For each $m\in\{2,3,\ldots,n\}$, let $CT_m$ be the group generated by all horizontal class transpositions $\tau_{r_1(m),r_2(m)}$, and define

$$CT_{(n)}=\langle CT_2,CT_3,\ldots,CT_n\rangle.$$

Let $N=\operatorname{lcm}(2,3,\ldots,n)$. Bardakov–Iskra conjecture. For $n>3$, the group $CT_{(n)}$ is isomorphic to the symmetric group $S_N$.

This conjecture concerns the finite groups generated by horizontal class transpositions and was proposed by Bardakov and Iskra; it is recorded as Problem 21.8 in the cited source. Its resolution is not established in the supplied text.