The non-homaloidality conjecture for the generic Hankel determinant

Let $R$ be the polynomial ring containing the entries of the generic $m\times m$ Hankel matrix $\mathcal{H}_m$. Hankel non-homaloidality conjecture. The determinant of $\mathcal{H}_m$ is not homaloidal.

A homogeneous polynomial is homaloidal when its polar, or gradient, rational map is birational onto its image; thus the conjecture predicts that the Hankel determinant does not have this property. The source explains that an affirmative answer to the preceding conjecture would imply this claim via a stated criterion, and that the case $m=3$ is known affirmatively, while the general case is left open.