Failure of the Multiplicative Rule of $k$ for linear factors

Let $R=\mathcal{A}/I$ be a quotient ring, and let $g_S$ and $h_S$ denote the products associated with a subset $S$ in the definition of the Multiplicative Rule of $k$. Failure of the Multiplicative Rule of $k$. For any $k\ge0$ and $N>k$, the Multiplicative Rule of $k$ fails for $g_i=1+xu_i$ and $h_i=1+xv_i$, $i=1,2,\dots,N$. Thus, the commutation relations for all subsets cannot in general be equivalent to those for subsets of cardinality at most $k$. The claim concerns the simplified linear-factor setting and is stated without a supplied resolution status.