Shalev's Engel word-map conjecture

Let $G$ be a finite simple non-abelian group, and let $n\geq 1$. The $n$-th Engel word is the word $e_n=[y,[y,\ldots,[y,x]\ldots]]$ with $y$ occurring $n$ times, and its word map is

$$G\times G\longrightarrow G.$$

Shalev's conjecture. For every finite simple non-abelian group $G$ and every $n\geq 1$, the $n$-th Engel word map

$$G\times G\longrightarrow G$$

is surjective.

The conjecture extends the Ore conjecture from the commutator word to all Engel words. The supplied source gives no resolution status for Shalev's conjecture.