Homological Zariski Problem B for initial forms

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be reduced holomorphic function-germs, and let $f_*$ and $g_*$ be their initial forms. Suppose there is a homeomorphism $\varphi\colon(\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ such that $f=g\circ\varphi$. Homological Zariski Problem B conjecture. The spaces $f_*^{-1}(0)\setminus\{0\}$ and $g_*^{-1}(0)\setminus\{0\}$ should have the same homology. This is stated as a weaker form of Zariski's Problem B; the paper notes that a positive answer would imply the right-equivalence version of Zariski's multiplicity conjecture, but gives no general resolution.