The multiple monomial scheme conjecture

Let $X=\mathbb{P}^N\subset\mathbb{P}^{N+M}$ be a linear subspace of dimension $N\geq 6$. Let $W$ be a Cohen–Macaulay multiple monomial scheme with reduced subscheme $W_{\operatorname{red}}=X$, and let $Y$ be a Cohen–Macaulay multiple scheme satisfying

$$\mathcal{I}_W\mathcal{I}_X\subset\mathcal{I}_Y\subset\mathcal{I}_W$$

and $\operatorname{deg}Y=\operatorname{deg}W+2$. The multiple monomial scheme conjecture. There exists a Cohen–Macaulay scheme $Z$ of degree $\operatorname{deg}W+1$ such that

$$W\subset Z\subset Y.$$

The source presents this as a more general multiple-structure formulation equivalent to Hartshorne's conjecture; its resolution is not established by the supplied text.