Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane

The main goal of this work is to show that if two weighted homogeneous (but not homogeneous) function-germs $({\text{C}}^2,0)\to (\text{C},0)$ are bi-Lipschitz equivalent, in the sense that these function-germs can be included in a strongly bi-Lipschitz trivial family of weighted homogeneous function-germs, then they are analytically equivalent.