Normal-subgroup prime-power covering-set conjecture

Let $A$ be a finite group with normal subgroup $G$ of index $|A:G|=n$, and let $U\leqslant G$ satisfy $P_A(U)=P_A(G)$, where $P_A(V)$ is the union of the $A$-conjugacy classes of elements of prime-power order that meet $V$. Normal-subgroup prime-power covering-set conjecture. There is an increasing integer function $f$ such that

$$|G:U|<f(n).$$

In the normal-subgroup case, equality of the prime-power covering sets forces $U\leqslant G$, making this a special case of the preceding conjecture and closely related to an earlier conjecture cited in the source. Its status is not resolved in the supplied text.