On the Milnor fibres of initial forms of topologically equivalent holomorphic functions

Budur, Fernandes de Bobadilla, Le and Nguyen (2022) conjectured that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. In this note, we give affirmative answers to this conjecture in the case of plane curves. We show also that a positive answer to this conjecture implies in a positive answer to the famous Zariski multiplicity conjecture both in the case of right equivalence or in the case of hypersurfaces with isolated singularities.