Subgroup for which any join of conjugates is a join of finitely many conjugates

Definition
Suppose $$H$$ is a subgroup of a group $$G$$. We say that $$H$$ is a subgroup for which any join of conjugates is a join of finitely many conjugates if, for any subset $$S$$ of $$G$$, there is a finite subset $$T$$ of $$G$$ such that:

$$\langle \bigcup_{s \in S} sHs^{-1} \rangle = \langle \bigcup_{t \in T} tHt^{-1}\rangle$$