Ein–Lazarsfeld syzygy conjecture for Veronese varieties

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,q}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the corresponding Koszul cohomology group. Ein–Lazarsfeld conjecture. Fix an index $1\leq q\leq n$ and $d\geq n+1$. Then

$$\mathrm{K}_{p,q}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))\neq 0$$

if and only if

$$\binom{d+q}{q}-\binom{d-1}{q}-q\leq p\leq \binom{d+n}{n}-\binom{d+n-q}{n-q}+\binom{n}{n-q}-q-1.$$

The nonvanishing direction was proved by Ein and Lazarsfeld, while the vanishing direction remains the focus of the paper; the conjecture is known when $n=2$.