Gill–Pyber–Szabó product conjecture for normal subsets of finite simple groups

Let $G$ be a non-abelian finite simple group, and let $S_1,\ldots,S_k$ be normal subsets of $G$, meaning unions of conjugacy classes. Gill–Pyber–Szabó's product conjecture. There exists an absolute constant $c$ such that, whenever

$$\prod_{i=1}^{k}|S_i|\geq |G|^c,$$

then

$$S_1\cdots S_k=G.$$

The paper proves this conjecture, so the asserted universal product bound is established for all non-abelian finite simple groups.