Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form $$r e_{i_1} \wedge \cdots \wedge e_{i_k} \quad \textrm{where} \ x_{i_\ell} r \in I \ \textrm{for all} \ 1 \leq \ell \leq k ?$$ These elements are certainly contained in the Koszul homology. Moreover, this does constitute a generating set, for example, for stable ideals, since one can show that the Koszul homology is actually minimally generated by a subset of elements of the above form. I have computed a fair amount of examples and it seems true more generally that this is a generating set.

I'm not sure if this is well-known or perhaps false, and any help or references for this would be greatly appreciated.