Homeomorphism conjecture for Milnor fibers of ambient-homeomorphic hypersurface singularities

Let two hypersurface singularities be ambient-homeomorphic if their ambient spaces and singularities are related by an ambient homeomorphism, and let $F_f$ denote the Milnor fiber of a hypersurface singularity $f$. Homeomorphism conjecture. Two ambient-homeomorphic hypersurface singularities have homeomorphic Milnor fibers. The homotopy type of the Milnor fiber is a topological invariant, and in the setting of images of finitely determined germs the boundary of the Milnor fiber is topological. The conjecture asks whether the full Milnor fiber is determined up to homeomorphism; the source notes that this is known for isolated singularities but is unresolved in the non-isolated case.