Existence of a non-locally nilpotent Engel group

Let $G$ be an Engel group, meaning that there is an integer $n$ such that

$$[[x,\underbrace{y],\dots,y}_{n}]=1$$

for all $x,y\in G$. Engel-group conjecture. There exists an Engel group that is not locally nilpotent. This is a long-standing open problem; in the setting of $PI$-groups, every Engel group is locally nilpotent, but it is not known whether an arbitrary non-locally nilpotent Engel group exists.