Relationship among induced covering-number boundedness notions

Determine whether there is a deeper relationship governing the boundedness and separation behavior of induced union-, local-, and folded-covering numbers, particularly whether a boundedness or separation result for one of these covering numbers systematically implies the corresponding result for the others under specified density conditions.

Background

The induced covering numbers satisfy the hierarchy $\icn{f} \leq \icn{l} \leq \icn{u} \leq \icn{g}$. The paper observes that, for several density restrictions, union, local, and folded variants exhibit similar boundedness and separation behavior, but it does not establish a general theorem explaining this pattern. The authors explicitly ask whether a deeper relationship exists.