Groups of Class Transpositions with Prescribed Prime Divisors of the Moduli

For a set $\mathcal{P}$ of odd primes, let ${\mathrm{CT}}_{\mathcal{P}}(\mathbb{Z})$ denote the group generated by all class transpositions whose moduli have no odd prime divisors outside $\mathcal{P}$. We prove that $$⟨{\mathrm{CT}}_{{\mathcal{P}}_1}(\mathbb{Z}),{\mathrm{CT}}_{{\mathcal{P}}_2}(\mathbb{Z})⟩={\mathrm{CT}}_{{\mathcal{P}}_1\cup {\mathcal{P}}_2}(\mathbb{Z})$$ for any sets ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ of odd primes. This gives a negative answer to Question 21.75 in the Kourovka Notebook.