The Hankel gradient ideal colon conjecture

Let $k$ be the base field, let $R$ be the polynomial ring containing the entries of the generic $m\times m$ Hankel matrix $\mathcal{H}_m$, and let $J$ be the gradient ideal of $\det(\mathcal{H}_m)$. Write $P=I_{m-1}(\mathcal{H}_m)$ for the ideal of its submaximal minors. Hankel gradient ideal colon conjecture. For every integer $i$ with $0\leq i\leq m-2$, one has

$$JP^i:P^{i+1}=I_{m-2-i}(\mathcal{H}_m).$$

This conjecture predicts the successive colon ideals governing the powers of the submaximal-minor ideal. Its stated consequences include that $I_{m-2}(\mathcal{H}_m)$ is the only other, necessarily embedded, associated prime of $R/J$, that the reduction number of $P$ with respect to $J$ is $m-2$, and that the partial derivatives of the determinant are algebraically independent. The source notes that the case $m=3$ is proved, while the general statement is posed as an open conjecture.