Solvability conjecture for conjugacy-class powers yielding inverse classes

Let $G$ be a group and let $K$ be a conjugacy class. Let $n\in\mathbb{N}$ with $n\geq 2$, and let $D$ be a conjugacy class. Write $D^{-1}$ for the inverse conjugacy class and $K^n$ for the $n$-fold product of $K$.

Solvability conjecture for inverse-class products. If

$$K^n=D\cup D^{-1},$$

then $\langle K\rangle$ is solvable. In particular, $G$ is not simple.

This conjecture concerns the case in which a power of one conjugacy class is the union of two mutually inverse conjugacy classes. The paper studies such products using structural and character-theoretic methods; whether the stated solvability conclusion holds in full generality remains open.