Let $G$ be a finite group. Denote $$G^1 = G, G^2 = [G,G], G^3 = [G,G,G] = [[G,G],G], \ldots, G^k = [G^{k-1},G].$$
Suppose we consider a commutators of $n$ copies of $G$ that are not neccesarily left associated. For example if $n=5$, we might have something like $[[G,G], [G,G], G]$. We refer to such an object as a commutator of weight $5$.
The following result is well-known.
Theorem. Every weight $n$ commutator of copies of $G$ is contained in $G^n$.
Now let us generalise this situation and for arbitrary subgroups $A_1,\ldots A_n$ we define analogously the commutator of weight $n$. For example for $n=5$ we can consider $[[A_1,A_2],[A_3,A_4,A_5]]$ or $[[A_1,A_2],[A_3,A_4],A_5]$ .
My question. Is it true again that every such commutator of weight $n$ belongs to the left-associated commutator $[A_1,A_2,\ldots,A_n]$?
I suspect that in general the answer is no. So I am also interesting when the answer is yes.