The link and double-point image criterion for bi-Lipschitz equivalence of map germs

Let $\mathbb{K}$ be the real or complex numbers, and let $f,g:(\mathbb{K}^n,0)\to(\mathbb{K}^p,0)$ be finitely determined map germs with $2n-1\leq p$. Write $link(f)$ and $link(g)$ for their links, and $D(f)$ and $D(g)$ for their double point sets. In the complex case, also let $\mathcal{L}(f)$ and $\mathcal{L}(g)$ denote their complex links.

Link and double-point image criterion. The map germs $f$ and $g$ are bi-Lipschitz equivalent if and only if $link(f)$ and $link(g)$ are bi-Lipschitz equivalent and the images $f(D(f))$ and $g(D(g))$ of their double point sets are bi-Lipschitz equivalent. In the complex situation, the considered link may be either the real link or the complex link $\mathcal{L}(f)$, respectively $\mathcal{L}(g)$.

This proposes that, in the range $2n-1\leq p$, the link together with the bi-Lipschitz type of the image of the double point set completely determines the bi-Lipschitz equivalence class of a finitely determined map germ. The source presents it as a conjectural criterion; no resolution is supplied here.