Orders of products of horizontal class transpositions

The group $\operatorname{CT}(\mathbb{Z})$ of class transpositions was introduced by S. Kohl in 2010. This is a countable subgroup of the group $\operatorname{Sym}(\mathbb{Z})$ of all permutations on the set $\mathbb{Z}$ of integers. We study products of two class transpositions in $\operatorname{CT}(\mathbb{Z})$ and give a partial answer to Kohl's Question 18.48 in The Kourovka Notebook. We introduce the set of horizontal class transpositions and prove that the order of the product of two horizontal class transpositions belongs to the set $\{1,2,3,4,6,12\}$ and any number in this set is the order of the product of some pair of horizontal class transpositions.