Arad–Herzog conjecture on products of non-trivial conjugacy classes

Let $G$ be a non-abelian finite simple group. A non-trivial conjugacy class is a conjugacy class other than the identity class. For $n\in\mathbb{N}$ with $n\geq 2$, consider the product of $n$ non-trivial conjugacy classes of $G$.

Arad–Herzog conjecture. This product is not a conjugacy class.

The conjecture generalizes the original assertion of Z. Arad and M. Herzog for the product of two classes. It has been confirmed for several families of finite simple groups, including sporadic simple groups, and for squares of a conjugacy class; the general case remains open.