Finite-conjugate-join-closed for a subgroup property in a group satisfying ascending chain condition on that subgroup property implies conjugate-join-closed for that subgroup property

Statement
Suppose:


 * $$\alpha$$ is a subgroup property.
 * $$G$$ is a group in which there is no infinite ascending chain of subgroups all satisfying $$\alpha$$ in $$G$$, i.e., $$G$$ satisfies the ascending chain condition on subgroups with property $$\alpha$$.
 * $$H$$ is a subgroup of $$G$$ with the property that any join of finitely many conjugate subgroups of $$H$$ in $$G$$, each of which satisfies $$\alpha$$, must also satisfy $$\alpha$$.

Then, the join of arbitrarily many (i.e., possibly infinitely many) conjugates of $$H$$ in $$G$$ also satisfies property $$\alpha$$.