Zariski's multiplicity conjecture

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $V(f)$ and $V(g)$ denote their zero-set germs at the origin. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,V(f),0)\to(\mathbb{C}^n,V(g),0)$. Zariski's multiplicity conjecture. Then

$$m(V(f),0)=m(V(g),0).$$

This conjecture remains open even for germs with isolated singularities, although the corresponding result for families with isolated singularities was confirmed by Fernández de Bobadilla and Pełka.