For a finite $p$-group $G$, let $\mho_i(G)$ denote the subgroup generated by $p^i$-powers of elements of $G$.

It is well-known that $\mho_i(\mho_j(G))$ can differ from $\mho_j(\mho_i(G))$ and from $\mho_{i+j}(G)$; for example, in the groups of order $2^8$ and ID between $515$ and $524$ have $\mho_1(\mho_2(G))=\mho_3(G)=1$ but $\mho_2(\mho_1(G))=C_2$.

However, it seems that $\mho_i(\mho_j(G))\subseteq\mho_j(\mho_i(G))$ whenever $i\le j$.

I have not been able to find this in the literature, nor do I have a proof.