  L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a
profinite group such that the set of solutions of the equation $x^n=1$ has
positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$
such that all elements of the coset $tH$ have order dividing $n$ (see Problem
14.53 of [The Kourovka Notebook, No. 19, 2019]). The validity of the conjecture
has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we study
the conjecture for compact groups $G$ which are not necessarily profinite and
$n=3$; we show that in the latter case the group $G$ contains an open normal
$2$-Engel subgroup.
