Equivalence conjecture for weighted homogeneous hypersurface germs and vector fields

Let $f\in\mathcal{O}_{(\mathbb{C}^{n+1},0)}$ define the isolated hypersurface germ $(V,0)=\{f=0\}$, and let $L_V$ be its link. A vector field on $(V,0)$ is transversal to the link if it is everywhere transversal to $L_V$. A vector field has a non-degenerate isolated singularity at $0$ when it extends to a neighborhood of $0$ in the ambient space and has a non-degenerate isolated singularity there.

Equivalence conjecture. The following statements are equivalent:

• $(V,0)$ is a weighted homogeneous germ.
• There exists a holomorphic vector field on $(V,0)$ that is everywhere transversal to the link $L_V$.
• There exists a holomorphic vector field that extends to a neighborhood of the origin in the ambient space with a non-degenerate isolated singularity at $0$.

The conjecture characterizes weighted homogeneity through the existence of suitable holomorphic vector fields. The source states that it is proved in part; the remaining implications or cases are not resolved in the supplied text.