Prime-power covering-set index conjecture

For a finite group $A$ and a subgroup $U\leqslant A$, let $\mathcal{P}(U)$ be the set of elements of $U$ of prime-power order, and define

$$P_A(U)=\{x^a:x\in\mathcal{P}(U),\ a\in A\},$$

the union of the $A$-conjugacy classes of prime-power-order elements meeting $U$. Prime-power covering-set index conjecture. There is an increasing integer function $g$ such that, whenever $U$ and $G$ are subgroups of $A$ with $P_A(U)=P_A(G)$ and $|A:G|=n$, one has

$$|A:U|<g(n).$$

The conjecture asks how far apart the indices of subgroups with the same prime-power covering set can be. The source gives examples showing that such subgroups need not have the same order, and records no resolution of the proposed uniform bound.