Hussain–Ma–Yau–Zuo's non-existence conjecture for negative derivations

Let $f\in\mathbb{C}[x_1,\ldots,x_s]$ be a weighted homogeneous polynomial defining a hypersurface with isolated singularity. Write $\mathcal{J}_n(f)$ for the ideal generated by the maximal minors of the order-$n$ higher Jacobian matrix of $f$, and let $w=(w_1,\ldots,w_s)\in\mathbb{N}_{\geq1}^s$ be the weight vector. Assume

$$\deg_w(f)\geq 2w_1\geq2w_2\geq\cdots\geq2w_s>0.$$

Hussain–Ma–Yau–Zuo's conjecture. The graded algebra

$$\mathbb{C}[x_1,\ldots,x_s]/\langle f,\mathcal{J}_n(f)\rangle$$

has no negative weighted derivations for $n\geq2$. The conjecture concerns the higher Nash blowup local algebra and is inspired by Halperin's question on negative derivations of local Artinian graded algebras. The source paper proves this assertion, while earlier work established cases including $s=n=2$ and $s=2$ with $n\geq2$.