Huppert's conjecture on character degrees

Prove that if L is a nonabelian finite simple group and a finite group G has the same set of complex irreducible character degrees as L, then G is isomorphic to the direct product of L and an abelian group A.

Background

Huppert's conjecture proposes recognition of a nonabelian finite simple group from the set of degrees of its complex irreducible characters, allowing an abelian direct factor in the competing group. The conjecture was posed by Huppert in 2000 and recorded in the Kourovka Notebook.

According to the paper, the conjecture had been verified for all finite simple groups except classical groups.