The generalized exponent conjecture for finite non-abelian simple groups

Let $G$ be a finite non-abelian simple group. Its generalized exponent is the least positive integer $n$ such that $(x^G)^n=G$ for every $x\in G$, where $x^G$ denotes the conjugacy class of $x$.

Generalized exponent conjecture. The generalized exponent of $G$ is at most $3$.

Many finite non-abelian simple groups are known to have generalized exponent at most $3$, and results of Shalev show that for a randomly chosen element the probability that $(x^G)^3=G$ tends to $1$ as $|G|\to\infty$. The conjecture asserts the uniform bound for every element of every finite non-abelian simple group.