The power-to-enhanced-power cograph conjecture for finite groups

Let $G$ be a finite group. Its power graph $\mathcal{P}(G)$ has vertex set $G$, with two distinct elements adjacent when one is an integer power of the other. Its enhanced power graph $\mathcal{E}(G)$ has vertex set $G$, with two distinct elements adjacent when they lie in a common cyclic subgroup. A graph is a cograph if it has no induced path on four vertices.

Power-to-enhanced-power cograph conjecture. For every finite group $G$, if the power graph $\mathcal{P}(G)$ is a cograph, then the enhanced power graph $\mathcal{E}(G)$ is a cograph.

This conjecture proposes that the implication, known for finite simple groups, holds for all finite groups. Results on nilpotent groups and on groups whose maximal cyclic subgroups intersect trivially provide supporting examples, but the general implication remains open.