The normal product decomposition conjecture for finite simple groups

Let $G$ be a non-abelian finite simple group, and let $S_1,\dots,S_k$ be normal subsets of $G$. Normal product decomposition conjecture. There exists $c$ such that, whenever

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

one has

$$G=S_1\cdots S_k.$$

This is proposed as a weaker conjecture that could provide a significant staging post toward the ideal product decomposition conjecture; the source gives no resolution.