Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.
Comment: Accepted for publication by the Journal of Topology