Parity version of Zariski's multiplicity conjecture

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ such that $f=g\circ\varphi$. Parity multiplicity conjecture. Then

$$m(V(f),0)\equiv m(V(g),0)\pmod 2.$$

This is presented as another version of the Zariski multiplicity conjecture, and the supplied text gives no evidence that it has been solved or refuted.