Regular-sequence conjecture for the generic Hankel gradient ideal

Let $\mathcal H_m$ be the fully generic $m\times m$ Hankel matrix, let $J=J_m[0]$ be its gradient ideal, and let $x_{m+3},\ldots,x_{2m-1}$ be the indicated variables. Regular-sequence conjecture. The sequence

$$\{x_{m+3},\ldots,x_{2m-1}\}$$

is regular modulo $J$. This conjecture concerns the residual algebraic structure of the generic Hankel gradient ideal and is proposed alongside the linear-rank conjecture; the supplied text gives no resolution.