Generic rank equality for odd-order Hankel tensors

Let $m\geq 5$ be odd, and let $\mathcal{H}\in\operatorname{H}^m(\mathbb{C}^n)$ be a generic Hankel tensor. The generic Hankel rank conjecture. The answer to whether

$$\operatorname{rank}(\mathcal{H})=\operatorname{rank}_S(\mathcal{H})=\operatorname{brank}(\mathcal{H})=\operatorname{brank}_S(\mathcal{H})=\operatorname{rank}_V(\mathcal{H})$$

is yes. The equality is known for generic Hankel tensors of even order or order three, but the rank relations for generic odd-order Hankel tensors with $m\geq 5$ remain unresolved; the corresponding assertion for all Hankel tensors is false.