Hussain–Ma–Yau–Zuo's contact invariance conjecture for higher Nash blowup algebras

Let $f$ and $g$ be germs in the convergent power series ring $\mathbb C\\{\mathbf{x}\\}$, with $f(\mathbf{0})=g(\mathbf{0})=0$. Two germs are contact equivalent at the origin if one is obtained from the other by a local coordinate change and multiplication by a unit. For $n\in\mathbb N^*$, let $\mathcal{J}_n(f)$ denote the $n$-th Jacobian ideal of $f$. Hussain–Ma–Yau–Zuo's conjecture. If $f$ is contact equivalent to $g$ at $\mathbf{0}$, then

$$\mathbb C\\{\mathbf{x}\\}/_{(f)+\mathcal{J}_n(f)} \cong \mathbb C\\{\mathbf{x}\\}/_{(g)+\mathcal{J}_n(g)}$$

as $\mathbb C$-algebras for every $n\in\mathbb N^*$. The conjecture proposed invariance of higher Nash blowup algebras under contact equivalence; it was proved by Le–Yasuda, including the non-isolated case, and hence is solved.