The Engel-word surjectivity conjecture for finite simple groups

For $n\in\mathbb{N}$, define the Engel words recursively by

$$e_1(x,y)=[x,y]=xyx^{-1}y^{-1},\qquad e_n(x,y)=[e_{n-1}(x,y),y]\quad(n>1).$$

For a group $G$, let $e_n:G\times G\to G$ be the corresponding Engel word map.

Engel-word surjectivity conjecture. For every $n\in\mathbb{N}$, the map $e_n$ is surjective for every finite simple non-abelian group $G$.

The conjecture extends the question of whether nontrivial word maps are surjective on finite simple groups. The source records the commutator case as the Ore conjecture, which is solved, but does not establish the assertion for all Engel words and all finite simple groups.