Homological initial-form Milnor-fibre conjecture for isolated singularities

Let $f,g\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ be holomorphic function-germs with isolated singularities at the origin, 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 initial-form conjecture. The spaces $f_*^{-1}(1)$ and $g_*^{-1}(1)$ should have the same homology. This is presented as a weaker isolated-singularity version of the initial-form Milnor-fibre conjecture, and no resolution is supplied in the text.