Non-existence of negative weight derivations on Jacobian algebras

Let $S=\mathbb{K}[x_1,\cdots ,x_n]$ be a polynomial ring in $n$ variables with positive integer weights over a field $\mathbb{K}$ of characteristic zero. Let $f\in S$ be a weighted homogeneous polynomial such that the Jacobian algebra $$A=S/(\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_1},\cdots ,\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_n})$$ is a finite-dimensional $\mathbb{K}$-vector space. We prove that $A$ has no nonzero negative weight derivation, which confirms a conjecture of the third author. An immediate consequence of this conjecture is the micro-local characterization of quasi-homogeneous singularities.