The Jacobian-minor primaryness conjecture for ideals generated by quadrics

Let $J$ be an ideal generated by quadrics in a polynomial ring $S$, and let $r={\rm ht}(J)\geq 2$. Let $\Theta$ be the Jacobian matrix of $J$, let $I_r(\Theta)$ denote its ideal of $r$-minors, and let $\mathfrak m$ be the irrelevant maximal ideal of $S$. The Jacobian-minor primaryness conjecture. The following conditions are equivalent:

1. $I_r(\Theta)=\mathfrak m^r$.
2. $I_r(\Theta)$ is $\mathfrak m$-primary.

This conjecture generalizes the equivalence established in the paper for ideals generated by the $2$-minors of a $2\times n$ matrix of linear forms. Its status is not resolved in the supplied text.