Existence criterion for codimension-two monadic complexes on projective space

Let $k\geq 2$, and consider the complex

$$0\longrightarrow \mathcal O_{\mathbf P^k}(-2)^n\stackrel{\beta^{\vee}}\longrightarrow \mathcal O_{\mathbf P^k}(-1)^{2n+r}\stackrel{\alpha^{\vee}}\longrightarrow \mathcal O_{\mathbf P^k}^{n+r+1}.$$

Assume that $\beta$ is surjective and that $\alpha$ degenerates in codimension $2$; equivalently, the cokernel in degree $0$ is $\mathcal I_X(2n-(2n+r))=\mathcal I_X(-r)$ for a locally Cohen–Macaulay subscheme $X\subseteq\mathbf P^k$ of codimension $2$. Existence criterion. The complex exists if and only if $r\geq 0$ and

$$n\leq \binom{r+3-k}{2}.$$

This gives a proposed numerical classification of the complexes under consideration, extending the preceding discussion of monads and codimension-two degeneracy loci. The supplied text does not indicate whether the assertion has been proved or remains open.