Lévai–Pyber conjecture on cosets of bounded order in profinite groups

Let $G$ be a profinite group, and let $n$ be a positive integer. Suppose that the set of solutions of

$$x^n=1$$

has positive Haar measure. Lévai–Pyber conjecture. Then $G$ has an open subgroup $H$ and an element $t$ such that every element of the coset $tH$ has order dividing $n$. This conjecture concerns when a positive-measure word fiber in a profinite group contains a full coset of an open subgroup. It was proved for $n=2$, while the general case is presented here as open.