  Boij-S\"oderberg theory concerns resolutions of graded modules over a
polynomial ring over a field. Specifically Boij-S\"oderberg theory gives a
description of the cone of Betti diagrams for Cohen-Macaulay modules. Eisenbud
and Schreyer discovered a duality between the cone of Betti diagrams and the
cone of cohomology tables for vector bundles on projective space. In the dual
theory an important role is played by so called natural vector bundles $E$
which have the property that the cohomology of every twist of $E$ is
concentrated in a single degree. In [4], Eisenbud and Schreyer consider the
bi-graded theory on $\mathbb{P}^1 \times \mathbb{P}^1$ and conjecture that
natural vector bundles exist with prescribed Euler characteristic. The Euler
characterist depends on three rational number $\alpha,\beta, \gamma$. We prove
this conjecture provided that $\alpha,\beta$ are not both integral.
