Thompson's conjecture on conjugacy class sizes

Prove that if L is a nonabelian finite simple group and a finite group G has the same set of conjugacy class sizes as L and has trivial center, then G is isomorphic to L.

Background

Thompson's conjecture concerns recognition of a nonabelian finite simple group from its conjugacy class-size set together with the condition that the competing group has trivial center. It was posed in the same context as Shi's conjecture and recorded in the Kourovka Notebook.

The paper states that the conjecture is known for all finite simple groups except the alternating groups, with the alternating-group case reduced to the binary Goldbach conjecture.