Thompson's conjecture on the existence in simple groups of a conjugacy class whose square is the whole group

Statement
In a finite simple non-abelian group, there exists a conjugacy class such that every element of the group can be expressed as a product of two elements from that conjugacy class.

Weaker conjectures

 * Ore's conjecture is a weaker version that states that every element in a finite simple non-abelian group is a commutator.