The linear-type conjecture for the Hankel determinant gradient ideal

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 its determinant. Linear-type conjecture. The ideal $J$ is of linear type.

This asks whether the defining equations of the symmetric algebra of the Hankel determinant's gradient ideal are generated by the linear syzygies. The source lists this among three open questions and reports that all the preceding conjectures are solved affirmatively when $m=3$, but leaves the general case open.