Every countable torsion-free group is a subgroup of a 2-generated torsion-free group with two conjugacy classes

Statement
Suppose $$G$$ is a countable torsion-free group. Then, there exists a group $$L$$ containing $$G$$ such that $$L$$ is 2-generated and is also a torsion-free group with two conjugacy classes.

Related facts

 * Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes

Original proof

 * Small cancellations over relatively hyperbolic groups and embedding theorems by Denis V. Osin