Conjugacy-distinguished element

Definition
An element $$g$$ in a group $$G$$ is termed conjugacy-distinguished if whenever $$h$$ is an element of $$G$$ not conjugate to $$g$$, there exists a normal subgroup of finite index $$N \le G$$, such that the images of $$g$$ and $$h$$ under the quotient map $$G \to G/N$$ are not conjugate in $$G/N$$.

A group where every element is conjugacy-distinguished is termed a conjugacy-separable group.