IMC 2024 Problem 4

Let $g$ and $h$ be two distinct elements of a group $G$ , and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$ . Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$ . Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$ ).