The set of $2$-genereted $C^*$-simple relatively free groups has the cardinality of the continuum

In this paper we prove that the set of non-isomorphic $2$-generated $C^*$-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of $C^*$-simple groups without free subgroups of rank $2$ was posed by de la Harpe in 2007.