Eisenbud and Schreyer's natural vector bundle conjecture on the quadric surface

Let $Q=P^1\times\mathbb{P}^1$ and write $E(a,b)=E\otimes\mathcal{O}_Q(a,b)$. A vector bundle $E$ has natural cohomology if, for every pair of integers $(a,b)$, the cohomology $H^*(E(a,b))$ is concentrated in one degree. For rational numbers $\alpha,\beta,\gamma$, set

$$p(x,y)=(x+\alpha)(y+\beta)-\gamma.$$

Eisenbud and Schreyer's conjecture. For any $p(x,y)\in\mathbb{Q}[x,y]$ with $\gamma>0$, there exists a vector bundle $E$ with natural cohomology and Hilbert polynomial

$$\chi(E(a,b))=\operatorname{rank}(E)p(a,b)$$

for $\operatorname{rank}(E)$ sufficiently big. The paper proves this conjecture when $\alpha$ and $\beta$ are not both integral; the remaining cases are not resolved here.