Isolated-singularity version of Zariski's multiplicity conjecture

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be holomorphic function-germs with isolated singularities at the origin. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,V(f),0)\to(\mathbb{C}^n,V(g),0)$. Isolated-singularity multiplicity conjecture. Then

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

The paper states that the general Zariski multiplicity problem remains open even in this isolated-singularity setting; only a result for families with isolated singularities is reported as confirmed.