Liebeck–Nikolov–Shalev conjecture on products of subset conjugates

Let $G$ be a finite simple group and let $A\subseteq G$ be a subset with $|A|\geq 2$. A conjugate of $A$ means a set $gAg^{-1}$ for some $g\in G$. Liebeck–Nikolov–Shalev conjecture. There exists an absolute constant $N$ such that $G$ is the product of at most

$$N\frac{\log|G|}{\log|A|}$$

conjugates of $A$. The conjecture concerns efficient product decompositions in finite simple groups; in the source, its proof is stated to have been completed by Lifshitz, so the conjecture is no longer open.