Boij-Soderberg theory for ideals generated by degree 2

In Boij-Soderberg theory, it is known that for any degree sequence $\mathbf{d}$, there exists a finitely generated module that has a pure resolution of type $\mathbf{d}$. On the other hand, in the case of ideal, there are two necessary conditions for the degree sequence, which $\mathbf{d}$ satisfies them if there is an ideal that has a pure resolution of type $\mathbf{d}$. In this paper, by theory of generic initial ideals and Boij-Soderberg decompositions, we construct the degree sequence which satisfies these conditions but there is no such an ideal.