Simon’s conjecture for monomial ideals

Let $S$ be a polynomial ring, and let $I\subset S$ be a monomial ideal. Denote its minimal monomial generating set by $G(I)$. Assume that $I$ has linear quotients, meaning that its minimal generators can be ordered so that every successive colon ideal is generated by variables. For monomial ideals $I\subset J$ with $G(I)\subset G(J)$, say that $I$ can be extended to $J$ by linear quotients if the generators in $G(J)\setminus G(I)$ can be ordered so that every successive colon ideal is generated by variables.

Simon’s conjecture for monomial ideals. If $G(I)\subset G(\mathfrak{m}^d)$, then $I$ can be extended to $\mathfrak{m}^d$ by linear quotients.

This is the proposed extension of Simon’s squarefree conjecture from the squarefree Veronese ideal to the full power of the homogeneous maximal ideal. The source gives no resolution status for this extension.