Let $G$ be a group and let $R_1,\cdots, R_n$ be subgroups of $G$, where $n\geq 2$. The symmetric commutator product of $R_1,\cdots, R_n$, denoted by $[R_1,\cdots,R_n]_S$, is defined as $$[R_1,\cdots,R_n]_S:=\displaystyle\prod_{\sigma\in\Sigma_n}[\cdots[R_{\sigma(1)},R_{\sigma(2)}],\cdots,R_{\sigma(n)}]\text{,}$$ where $\Sigma_n$ is the symmetric group of degree $n$.
For each $g\in G$, let $g^G$ be the normal closure of $g$ in $G$; that is, $g^G$ is the smallest (by inclusion) normal subgroup which contains $g$.
Let $x_1,\cdots,x_n\in G$ and $N$ be a normal subgroup of $G$. Suppose that all iterated commutators $$[\cdots[x_{\sigma_1},x_{\sigma_2}],\cdots,x_{\sigma_n}]\in N\text{,}$$ where $\sigma\in\Sigma_n$. Is it true that $$[x_1^G,\cdots,x_n^G]_S\subseteq N\text{?}$$