Failure of the group-theoretic Rule of $k$

Fix integers $k\ge0$ and $N>k$. Let $g_1,\dots,g_N,h_1,\dots,h_N$ be generators of a group subject to the relations $g_Sh_S=h_Sg_S$ for all subsets $S$ with $|S|\le k$. Failure of the group-theoretic Rule of $k$. Then $g_Sh_S\ne h_Sg_S$ for any subset $S$ with $|S|>k$. This statement is presented as a consequence that would follow from the preceding multiplicative conjecture; the supplied text does not give a resolution.