The autocorrect-sequence conjecture for products of isomorphic simple groups

Let $\overrightarrow{u}$ be an autocorrect sequence, and let a $\overrightarrow{u}$-Engel automorphism mean an automorphism satisfying the corresponding Engel condition. Consider a group $G$ of the form

$$G=\prod G_i,$$

where all the groups $G_i$ are isomorphic simple non-abelian groups. Product-group autocorrect-sequence conjecture. There exists an autocorrect sequence $\overrightarrow{u}$ such that no such group $G$ has a non-trivial $\overrightarrow{u}$-Engel automorphism. The paper later states a theorem asserting equivalence with the finite-group solvable-radical conjecture; the supplied text gives no resolution of either conjecture.