The substantial unfolding characterization conjecture for quasi-homogeneous map-germs

Let $f\colon(\mathbb{C}^n,0)\to(\mathbb{C}^p,0)$ be an $\mathcal{A}$-finite map-germ whose stable unfolding with the minimal number of parameters lies in the nice dimensions. A stable unfolding $F\colon(\mathbb{C}^n\times\mathbb{C}^r,0)\to(\mathbb{C}^p\times\mathbb{C}^r,0)$ is substantial if, for a parameter coordinate $\Lambda$, one has $\Lambda\in d\Lambda(\operatorname{Lift}(F))$, where $\operatorname{Lift}(F)$ consists of target vector fields liftable over $F$. Substantial unfolding characterization conjecture. The map-germ $f$ is quasi-homogeneous if and only if it admits a substantial unfolding. The paper presents this as a proposed conjecture; no resolution status is given.