A note on Engel elements in the first Grigorchuk group

Let $\mathrm{\Gamma }$ be the first Grigorchuk group. According to a result of Bartholdi, the only left Engel elements of $\mathrm{\Gamma }$ are the involutions. This implies that the set of left Engel elements of $\mathrm{\Gamma }$ is not a subgroup. Of particular interest is to wonder whether this happens also for the sets of bounded left Engel elements, right Engel elements, and bounded right Engel elements of $\mathrm{\Gamma }$. Motivated by this, we prove that these three subsets of $\mathrm{\Gamma }$ coincide with the identity subgroup.