Discreteness of c-smooth conjugacy classes

Determine whether every c-smooth conjugacy class in a pro-oligomorphic group is discrete; equivalently, prove that each element of a c-smooth conjugacy class is an isolated point of that conjugacy class in the group topology, or provide a counterexample.

Background

In the paper, an element g of a pro-oligomorphic group G is called c-smooth if its centralizer Z(g) is an open subgroup. Such elements form a normal subgroup G_sm, and c-smoothness captures smoothness with respect to the conjugation action.

Proposition 3.6 shows that isolated points in conjugacy sets are c-smooth, and several examples demonstrate when isolated points occur. The authors pose the natural converse problem of whether c-smoothness forces discreteness (i.e., all points are isolated) of the entire conjugacy class.