Eisenbud's low-rank quadric syzygy conjecture

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $S$ be a polynomial ring over $\Bbbk$, and let $I\subseteq S$ be a prime ideal containing no linear form. Suppose that $I_2$ is spanned by quadrics of rank at most four, and write $2LP(S/I)=n$. Eisenbud's conjecture. The ideal $I$ contains the $2\times 2$ minors of a 1-generic $p\times q$ matrix, where

$$p+q-3=n.$$

This predicts that low-rank quadratic generators force the determinantal structure arising from 1-generic matrices. The supplied text gives no resolution status.