The ideal product decomposition conjecture for finite simple groups

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

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

there exist elements $g_1,\dots,g_k$ such that

$$G=(S_1)^{g_1}\cdots(S_k)^{g_k}.$$

This conjecture would significantly generalize the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev, and the authors state that it seems out of reach at present.