Assume that I have a reduced zero-dimensional scheme $Z \subset \mathbb{P}^3$ of degree $mn$ having the following property:

For a general outer projection $\pi \colon \mathbb{P}^3 \to \mathbb{P}^2$, the subset $Z':=\pi(Z) \subset \mathbb{P}^2$ is a complete intersection of bidegree $(m,n)$, in other words the resolution of its ideal sheaf $\mathcal{I}_{Z'}$ is the Koszul complex $$0 \to \mathcal{O}_{\mathbb{P}^2}(-m-n) \to \mathcal{O}_{\mathbb{P}^2}(-m) \oplus \mathcal{O}_{\mathbb{P}^2}(-n) \to \mathcal{I}_{Z'} \to 0.$$

My question now is:

Which kind of information does this give about the syzygies of the ideal sheaf $\mathcal{I}_Z$ in $\mathbb{P}^3$? For instance, I would like to know whether there is a generator in degree $2$, i.e. whether $Z$ is contained in (at least one) quadric surface.

This arises as an attempt to study the problem stated in my previous question When is a general projection of $d^2$ points in $\mathbb{P}^3$ a complete intersection? (here $m=n=d$) which, despite the nice example provided by Dmitri Panov, is still unanswered in its general form.

Any answer or reference to the existing literature will be appreciated. Thank you.