The higher infinitesimal-neighbourhood conjecture for codimension-two linear spaces

Let $\mathbb{P}^N$ with $N\geq 6$ be a linear subspace of codimension two in $\mathbb{P}^{N+2}$. Let $M$ denote the integer occurring in the infinitesimal-neighbourhood notation, and let $Y$ be a Cohen–Macaulay scheme satisfying

$$(\mathbb{P}^N)^{(M)}\subset Y\subset (\mathbb{P}^N)^{(M+1)}$$

and having degree equal to the degree of $(\mathbb{P}^N)^{(M)}+2$. The higher infinitesimal-neighbourhood conjecture. There exists a Cohen–Macaulay scheme $Z$ of degree equal to the degree of $(\mathbb{P}^N)^{(M)}+1$ such that

$$(\mathbb{P}^N)^{(M)}\subset Z\subset Y.$$

The source states that this multiple-structure conjecture is equivalent to Hartshorne's conjecture; its resolution is not established by the supplied text.