UL-intersection-closed subgroup property

Definition
A subgroup property $$p$$ is termed UL-intersection-closed if for any group $$G$$, any nonempty (but otherwise arbitrary, possibly infinite) indexing set $$I$$, and subgroups $$H_i \le K_i \le G, i \in I$$, such that $$H_i$$ satisfies property $$p$$ in $$K_i$$ for each $$i \in I$$, we have:

$$\bigcap_{i \in I} H_i$$ satisfies property $$p$$ in $$\bigcap_{i \in I} K_i$$.