Flatness conjecture for matrix Hessenberg degenerations over arbitrary sheets

Let $h:[n]\to[n]$ be a Hessenberg function, let $\mathsf{x}\in\mathfrak{gl}_n(\mathbb{C})$, and let $\varphi_{\mathsf{x},h}$ be the one-parameter family constructed from the associated curve in the sheet containing $\mathsf{x}$, with general fiber the matrix Hessenberg scheme $\mathcal{Y}_{\mathsf{x},h}$ and special fiber the associated nilpotent matrix Hessenberg scheme. Flatness conjecture. The morphism

$$\varphi_{\mathsf{x},h}$$

is flat for every $\mathsf{x}\in\mathfrak{gl}_n(\mathbb{C})$. The paper proves flatness for the minimal sheet and proposes this statement as the extension to every sheet; it would give flat degenerations of all matrix Hessenberg schemes to nilpotent ones.